To mechanics of deformation, flow, and fracture. S.L. Arsenjev 1. Physical-Technical Group Dobroljubova Street, 2, 29, Pavlograd Town, Ukraine

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1 To mehanis of deformation, flow, and frature S.L. Arsenjev 1 Physial-Tehnial Group Dobroljubova Street, 2, 29, Pavlograd Town, 514 Ukraine It is stated in the main in essene new approah to mehanis of the stressed state of the solid body from statistially isotropi material and the homogeneous liquid dynamis. The approah essene is in the deteted property of the ore-shell spontaneous struturization of internal energy of the solid and liquid bodies in its natural state and under ation of external fores. The method elements of onstrution of physially adequate model of the stressed state of the solid and liquid bodies, reprodued exatly its behavior on the stages of elasti and plasti deformation, flow and frature, are stated. It is addued a number of the examples of the stressed state onstrution of the simple form bodies under ation of its tension, ompression, torsion and at its ontat interation. For the first time it is addued struture of the prinipal normal - stresses in ylindrial bar under ation of the torsion moment. The deteted property and the developed method is one of neessary bases for onstrution of physially adequate mathematial model of the stressed state of the body and fluid in ontrast to traditional approah. PACS: b; b; a; g Introdution The mehanis analysis of a forming of the nek in ylindrial speimen from statistially isotropi and elasti-plasti material, in partiular, from the mild steel under ation of the longitudinal (axial) tension fores, stated in author s artile [1], had led to neessity to aeptane of two following suppositions: firstly, on distribution of trajetories of prinipal normal stresses in this speimen in the kind of ylindrial helies with its angle 45 and, seondly, on a self-sruturization of the speimen volume into its ore and shell parts. These two suppositions had allowed to aount rationally for all features of the stress and strain states of suh speimen at all stages of its testing, inluding its frature. Aording to the seond from the aepted suppositions, the averagestatistial quantity of diameter of the speimen ore part makes up 2 2 = of its external diameter. This ratio remains to be orret for the nek diameter in omparison with the speimen external diameter, and the same ratio is orret for the nek ore part in omparison with the nek diameter. Deviation from the indiated quantity not exeeds 2 % for the different sorts of strutural steel [2]. The addued ratio in ombination with the first from the aepted suppositions has allowed geometrially strit and physially adequate to onstrut the speimen form at all stages of its loading right up to its frature with use only two geometrial surfaes: the hollowed orthogonal right hyperboloid and evolventoid of rotation. Suh approah has allowed to asertain that the neking proess is a phenomenon of the loal straightening of helial trajetories of the prinipal stresses and, aordingly to it, of a transformation of retilinear generatrix on the speimen ylindrial surfae into a hyperbola in the very narrow zone of the speimen nek. Presentation of the ylindrial volume in the kind of a struture, ontained in itself the ore and shell parts, has allowed to disover the real three-dimensional struture of the stressed state in the ylindrial speimen under simple tension and in that way to show that the one-dimensional stressed state as only stress in three-dimensional body is an abstration, devoid of physial sense. Physially substantial and mathematially strit solution of a problem on simple tension of ylindrial speimen and some onsequenes, stated in previous artile [1], an be, in author s opinion, applied 1 Phone: ( ) 4596 (home Rus.) ptgleus@yahoo.om

2 for physially adequate solution of problems on the stressed-strained state of the speimens with different form of its ross-setion under not only the tension test, but also under axial ompression, torsion and some other kind of a testing under both stati and dynami load. The prinipal problem of a generalized approah to a problem solution on the stressed-strained state of the speimens under the numbered kinds of a testing onsists in possibility to eluidate nature and features of the spontaneous struturization of statistially isotropi material in a speimen volume before appliation to it a test load and the influene of this phenomenon onto the stress distribution in the speimen under ation of external fores. Approah The approah essene to the stated problem solution onsists in the following theses: - a solid body in its initial unloaded state keeps its form and its volume owing to its internal energy, ensuring a stationary onnetion of miro partiles, onstituting the solid body material; an intensity of this onnetion is haraterized by the linear modulus and the volumetri modulus of 2 3 elastiity in dimensionality kgf m kgf m m as a volumetri density of the internal onnetion energy of the body material; thus, the solid body is in the own stationary stressed state, keeping its initial form and volume in a lak of external influenes; - the stationary onnetion of the body material partiles has the essentially osillatory harater and thereby stipulates a pulsatory osillatory harater of the own initial stressed state of the body from the simplest spherial form with its radial and shear pulsations to a ombination of radial, longitudinal, torsion, flexural pulsations in the ylindrial form body and up to the more ompliated ombinations of the pulsations in the ompliated form bodies; the stress waves, moving to the body surfae and refrating from it, form the pulsatory system of initial stationary stresses, orresponding struturally to the body form; in the result of it the initial stressed state of the body is essentially three-dimensional aordingly to the body form and this state has a ylial harater with the osillation frequeny, determined by a soni wave veloity in the body material, by the osillation mode and by the body harateristi dimensions; - any mehanial or thermal influene from without onto the essentially three-dimensional body an be able only quantitatively to hange a orrelation of omponents of its own initial threedimensional stressed state. In the given artile as a solid body is onsidered the elasti-plasti body from statistially isotropi material, as far as suh body under influene of external fores demonstrates ability to: - elasti (reversible) deformation; - the vaany-disloation adaptation of its material mirostruture to its stressed state on the plasti (irreversible) deformation stage; - the loal and as a whole flow of suh body material; - a loss of ontinuity on its frature stage. A body from brittle material, possessing by its tension strength in 2 5 times less than its ompression strength, demonstrates only little part of the plasti body behavior when other things being equal. At the same time the well-known experimental researhes [3], arried out by T. Karman (1911), had showed that the ylindrial speimens from marble demonstrate the plasti deformation under ation of a ombination of axial ompression with lateral hydrostati pressure. Solution As regards a distribution of trajetories of prinipal stresses in the kind of two families of ylindrial left- and right-handed helies, its existene is, although indiretly, orroborated by Lueders lines on the ylindrial speimen surfae on a stage of its plasti deformation under ation of the axial tension fore. A question on a possibility of the spontaneous struturization of the initial stressed state in isotropi three-dimensional body is, in the given artile, onsidered on the examples of bodies of some simple forms. Spherial body. One of the basi modes of its initial spontaneous osillations is realized in the kind 2

3 of radial osillation. In this ase an interation of the own osillation waves, moving to the sphere surfae, with the refrated waves, moving to the sphere entre, results in that the near-walled layer of the sphere material will be osillated in opposite phase to the sphere entral part. In that way the sphere volume ontains its ore and shell parts. Suh osillatory system an be presented in the kind of two bodies onneted by spring. It is known [4] that, firstly, the free osillations of suh mehanial system are going on without displaement of the system mass entre and, seondly, both bodies move in opposite phases with the same frequeny. When mass of these bodies is unequal then its osillation amplitudes is inversely proportional to its mass. In partiular ase, when mass of these bodies is the same, its amplitudes will be also equaled. As applied to the spherial body, this partiular ase orresponds to the ontinuity ondition of its material in a ontat boundary of this body ore and shell parts. For the sphere isotropi material, radius, dividing it on its ore and shell parts, is determined by equality of volumes of these parts 3 whene V = π ; Vs = 2V = π Rs, 3 3 R R R s =1 3 2 = , where R is radius of the sphere ore part, dividing the sphere volume into two equal parts and R s is radius of the sphere. Aordingly, a thikness of the sphere shell part makes up approximately 2.6 % from the sphere radius ( R s ). In a proess of these stationary radial osillations, the stress in the sphere ore is hanged periodially during one half-period from the volumetri ompression to the volumetri tension. Aordingly to it the stresses in the sphere shell the hoop stresses in two perpendiular diretions and radial stress are also hanged periodially from a tension to a ompression. In that way, it should be supposed the ontinual spherial body as an aggregate, onsisting from two elastially onneted parts, when an analysis of the stressed-strained state of suh body under ation of external load is arried out. In a ase of rotational osillations, the sphere ore radius is determined by an equality of the inertia moments of its ore and shell parts. Cylindrial body of a finite length. One of the basi modes of its initial spontaneous osillations of suh body is also realized in the kind of radial osillations. Ating analogially to the previous ase and supposing an equality of mass (volume) of its ore and shell parts, one an determine radius of the ylinder ore part 2 2 V = π R L; Vb = 2V = π Rb L, whene R R b = 1 2 =.77..., where R is the ylinder radius, dividing its volume into two equal parts, i.e. radius of the ylinder ore part; R b and L are radius and length of the ylinder respetively. In this mode of initial own osillations, the ylinder shell is periodially ompressing its ore. In the result, the ore is lengthened and one is tensioning the shell in longitudinal diretion in one halfperiod of the osillations. The next one half-period of the own osillations, the ylinder ore is shorten and, dilating simultaneously in radial diretion, one is arousing the tension hoop stress and the ompression longitudinal stress in the ylinder shell. These own osillations of the ylinder ore and shell parts orrespond to the osillatory nature of the longitudinal and transversal onnetions of miro partiles, onstituting the given body material. In the result, geometrial (vetor) sum of the hoop and longitudinal stresses forms trajetories of the prinipal stresses in the kind of two families left- and right-hand ylindrial helies in the ylinder shell. Suh struture of the stressed state of ylindrial speimen determines an orientation of a motion of miro deformations

4 (inluding to the speimen external surfae), stipulated by the vaany-disloation adaptation of the speimen material struture on its plasti deformation stage (the yield point zone) at the speimen tension test. In that way, a going out of the miro deformations onto the speimen external surfae visualizes the trajetories of the prinipal stresses in the kind of Lueders lines. The other mode of the own initial spontaneous osillations of the finite length ylindrial speimen is the longitudinal osillations of its parts at its ends relatively its middle ross-setion. These parts are osillated symmetrially in the opposite phases. A variety of the previous mode of the initial spontaneous osillations of the finite length ylindrial speimen is the torsion osillations of its parts at its ends relatively its middle ross-setion. Both these parts are also osillated in the opposite phases. The latter two modes determine an origin of the nek mainly in the middle ross-setion of ylindrial speimen at its tension test. And the latter mode of the torsion osillations of the speimen parts at its ends relatively its middle ross-setion is also of additional interest. A feature of this mode is in that only one family of the prinipal stresses is loaded by the tension fore during one half-period of these osillations. The other family of the prinipal stresses is found to be unloaded in this half-period. During the next half-period only the other family of the prinipal stresses is loaded. In the other words: in the every one half-period is found to be ativated only one family of spiral helies as of trajetories of the prinipal stresses. The desribed feature determines a basi differene of the stressed state of ylindrial speimen under ation of the tension fore and under ation of the torsion moment. In the first of these ases both the left-handed and right-handed families of trajetories of the prinipal stresses are simultaneously loaded. In ontrast to it, in the seond of these ases only one family of trajetories of the prinipal stresses only the left-handed or only the right-handed is loaded aording to an ation diretion of the torsion moment. Comparing these ases, one an understand that the hoop tension stresses in the shell part of the tensioned ylindrial speimen and radial pressure of its shell part onto its ore part will found to be twie as muh than in the same speimen under its torsion under ondition of equality of the prinipal stresses in its quantity. Aordingly, the lengthening of the speimen ore part under its torsion will found to be half in omparison with its tension under the above-mentioned ondition. Side by side with the own spontaneous torsion osillations, the ylindrial speimen an also experiene the rotational osillations of its shell part relatively its ore part around its longitudinal axis without twisting. In both these ases a diameter of the speimen ore part is determined by an equality of the polar moments of inertia of its ore and shell parts 4 i.e. whene J p = J ps ; or J p = 2J p, D D d π = 2π D d = D D,, where D, d are diameters of the speimen and its ore part orrespondingly. The numbered modes of the own spontaneous osillations of ylindrial body are the most atual for an analysis of the stressed-strained state of ylindrial speimen at its tension and torsion tests, although suh body an also experiene the other modes of its own spontaneous osillations. The onsidered example of a body of ylindrial form as one from the simplest geometrial forms and the same time as a form, used very widely in the mahine elements, allows, in author s opinion, to solve physially adequate the question on the stressed-strained state mehanis of the diverse form bodies under ation of external fores, using the desribed property of self- struturization orrespondingly to its form and, naturally, to its material homogeneity at its initial state.

5 5 Disussion of results Effetiveness of the stated approah to an analysis of the stressed-strained state of the solid body is further demonstrated by means of the well-known results of experiments, addued by A. Nadai in his book [5], and the some others, inluding own experiments of the given artile author. Mehanis of the neking in the tensioned ylindrial speimen from mild steel is, in general, desribed in the previous artile [1] of the given artile author. The neessary additional explanation is in the following. From the one hand, the diagonals of the lateral sides of geometrial body in a form of a ube - the so-alled external or lateral diagonals - irumsribe the linear surfae in the kind of the hollowed orthogonal (right) rotary hyperboloid at its rotation around entral axis of the ube. And from the other hand, the ube diagonals, passing from one of its apex to its opposite apex through its volume - the so-alled internal diagonals - are inlined to the ube bases under angle. Correspondingly to it the orthogonal hyperboloid generatries are inlined to its rotation axis and simultaneously to its bases under 45 angle, and the orthogonal hyperboloid diagonals, passing through its volume, i.e. its internal diagonals, are inlined to its rotation axis under = angle. Correspondingly, the helial trajetories of ation of the prinipal stresses on the external surfae of the tensioned ylindrial speimen are inlined to the speimen axis under 45 angle, the speimen shell frature is also realized under 45 angle, and the speimen ore frature is aompanied by a tearing on the series sites, inlined meanstatistially under ~ 55 angle to the speimen longitudinal axis. The above-mentioned angles are onneted between themselves by the following equality tan = sin45 = osine 45 = 1 2 = The addued explanation allows physially adequate to eluidate mehanis of the elasti and plasti deformation and frature of the speimens with non-irular ross-setion from mild metal at its tension test. The diversity of its ross-setion forms and of a orrelation of its diameters an be subdivided into the following ategories: - the speimens with a ompat (solid) ross-setion in the kind of a square, ellipse, retangle with ommensurable quantities of its sides or diameters; - the flat speimens; these speimens an be, in ones turn, subdivided into two following ategories: - the flat heavy-walled speimens with a orrelation of its ross-setion diameters or sides in the limits from 6 up to 1; - the flat thin-walled speimens with a orrelation of its ross-setion diameters or sides more than 1. Morphologial unity of the numbered ategories of the speimens is in that any retangular rosssetion an be formed by a trunating of a ylinder at least by two parallel eah other flat surfaes, plaed symmetrially along the ylinder longitudinal axis. In the result, the speimens with its ompat ross-setion are found to be lose by the ylindrial speimen, and the narrow rosssetion of the thin-walled speimens is symmetrially plaed relative the ylinder longitudinal axis and ontains in itself the internal diagonal of the ube, insribed into the ylinder. These geometrial features of the speimens with non-irular ross-setion determine a differene of its behavior at the tension test. Distintive feature of plasti deformation of the speimens with a ompat ross-setion is in that Lueders lines, inlined under angle 45 to the speimen axis, are plaed only in a stripe along the mean line of the most wide side of the speimen; in the speimen with the square ross-setion these stripes with Lueders lines is plaed on all its lateral sides. Suh loalization of Lueders lines testifies to existene of ylindrial part, insribed in the speimen volume and ontaining in oneself the ore and shell parts. Just owing to it a frature of suh speimen starts by a forming of the nek and omes to an end by a forming of the frature surfae in a form of the up and one, i.e. one is going on by the same way as one is going on at a frature of ylindrial speimen.

6 During a plasti deformation of the flat thin-walled speimen under ation of the tension fore Lueders lines are oriented under angle to the tension diretion. A frature of suh speimen starts by a forming of a loal thinning the linear nek, developing from lateral edge of one of its wide sides along the straight line, oinided with one of Lueders lines and, orrespondingly, with the ube internal diagonal. A frature omes to an end when the nek reahed the opposite edge of the speimen wide side. A form of a frature of the flat heavy-walled speimen is very sensitive to a quantity of a orrelation of its ross-setion sides (diameters) and one an be similar to one of the above-desribed two forms with symptoms of the other one. An example of suh behavior of the tensioned speimen is addued in [5]. A suitability of metal to its treatment by a pressing is determined by means of the axial ompression test of ylindrial speimens with initial orrelation of its height to its diameter h = h d about 1.5 at the normal and heightened temperature. The examples of plasti deformation and the flow struture of the ylindrial speimens from mild metals, obtained by means of its longitudinal utting, are also addued in the book [5]. Side by side with it V.G. Osipov [6] addued in fig. 4 the enlargement photo of a state of the ylindrial speimen, deformed plastially on 8 % of its initial height and ut aross. M.E. Drits and L.N. Moguthi [7] addued in its fig. 4 the natural sale photo of the outward appearane of the same plastially deformed speimen. Both photos, addued in [5, 6], show suffiiently learly a presene of three zones: - the entral irular zone of the metal grained non-oriented mirostruture by its diameter some lesser then the speimen initial diameter; - the hoop zone along the irumferene of the speimen external ontour with its internal diameter ~.74 of the external ontour diameter of the deformed speimen; the metal grained mirostruture in this zone is also non-oriented; three raks are developed from the speimen external ontour in longitudinal-radial diretion with its depth nearly on all width of the hoop zone; - the intermediate hoop zone with the learly visible radial orientation of the speimen metal mirostruture. These three zones are also learly visible on the butt-end flat surfae of the non-ut speimen [6], i.e. a presene of the self-struturing of ylindrial speimen is also orroborated by its axial ompression test. It should be noted that the intermediate hoop zone is formed at suffiiently high veloity of the uninterrupted axial ompression of the speimen not only up to its plasti deformation, but and to its flow. In the experiments, arried out by an initiative of the given artile author at suffiiently low veloity of the uninterrupted axial ompression, the ompression fore strain diagram had a ylial multi-stage harater. Every yle (stage) of this diagram was ontaining in itself two setions of the speimen deformation - elasti and plasti in the kind of the straight inlined line and the horizontal line respetively. The end of every suh yle was a beginning of the same next yle. In the given ase, for instane, this diagram was ontaining in itself four suh yles. In this test the intermediate hoop zone was not formed as in every yle as if the new speimen with its ross-setion area, inreased in the previous yle, partiipates in the every subsequent yle. At the high veloity of the uninterrupted axial ompression, when the deformation veloity exeeds veloity of the natural vaany-disloation adaptation of the speimen material stressed state, the ompression fore strain diagram is realized in the kind of only one yle. In this ase the initial plasti deformation of the speimen material turns into its radial flow, forming the intermediate hoop zone between the speimen ore and shell parts. Thesis on the initial spontaneous self-struturization of the stressed-strained state of a solid body aording to its form, elaborated in the author previous artile [1] and developed in the given artile, allows to produe mehanis of elasti and plasti deformation of ylindrial speimen from mild metal under ation of the axial ompression fore in the following way. The speimen parts, its ore and shell, having the same area of its ross-setion are equally ompressed by the axial pressure. At the same time a transversal radial deformation of the speimen ore part is restrited by the hoop stiffness of the speimen shell part. Beause of it the stressed state of the 6

7 speimen ore part is found to be lose by a hydrostati ompression. At the same time the speimen shell part is under ation of a ombination of the hoop tension stress, aroused by a pressing onto it of the speimen ore part, and of axial ompression. Both these stressed states promote to the plasti deformation of the speimen from mild metal. Naturally, even if partial ompensation of the hoop tension stresses in the speimen shell part by means of a loading it by lateral hydrostati pressure allows plastially to deform even suh brittle material as a marble [3]. On the other hand, a speimen from mild steel an be flattened out thus muh that under ation of radial pressure of the speimen ore part its shell part will be tore by the few radial-axial raks, developed from the speimen lateral ontour as it is showed in fig. 4 in the mentioned artile [6]. The quantitative valuation of the stressed state omponents of ylindrial speimen under ation of axial ompression orresponds, to a ertain degree, to the solution method, stated in the author artile [1], with the taking into aount of a reverse diretion of the ating fores at the speimen ompression. That is the solution omes to Lame problem for the heavy-walled ylinder, but already in the three-dimensional statement. In the ase of suffiiently quik axial ompression of ylindrial speimen from mild steel on a stage of its flow the solution of Lame problem, stated in [1], must be added by the expressions allowing taking into aount of the ontat frition and visosity of the speimen metal. Nadai in his book [8] has addued the referene (as a guide) quantities of dynamial visosity of some material, inluding a opper, strutural steel and the others. Speialists in the field of the metal treatment by a pressing P.I. Polukhin, V.A. Tjurin, P.I. Davidkov and D.N. Vitanov in their book [9] spare the separate hapter to the ontat frition of the metal half-produt with the press flat plates. The ontat frition fator distinguishes very suffiiently the ompression test from the tension test. The ontat frition restrits, to a ertain degree, radial deformation of the ompressed speimen and in that way one promotes to a forming of the supporting ones - in the ylindrial speimen volume - with its bases at the speimen butt-ends. The speimen material is in a state lose to a hydrostati ompression in a volume of these ones. In his book [8] Nadai addued the results, obtained by Meyer and Nehl (1925), of a measuring of angle ϕ between longitudinal axis of ylindrial speimen and the supporting one generatrix in dependene on the speimen relative height h = h d. In general, these results an be offered by a hyperbola, passing through points with oordinates: along the absissa axis h =., 1., 2.5 and along the ordinate axis ϕ = 9, 55, 32 respetively. One an see that when h = 1 then the inline angle of the one generatrix is equaled to the angle between the ube rotary axis and its internal diagonal as the ation trajetory of the prinipal stresses in the speimen volume. Axial ompression of the short speimens with h < 1 leads to a superposition of apexes of the supporting ones and, orrespondingly, to an inrease of the speimen resistane to the ompression fore, applied to the speimen. Axial ompression of the long speimen with h > 1 leads to a transformation of these ones into the ylindrial ore part of the speimen. S.P. Timoshenko [3] addued the results of experimental researh, arried out by C. Sahs (195), in the kind of the speimen relative height ( h ) ompressive stress diagram. An inrease of h in the limits from.27 up to 2. is aompanied by a derease approximately in two times of the ompression stress. These ylindrial speimens were made from mild opper of the same diameter. Axial ompression of the ylindrial speimens by the preliminary lubriated plates of the test mahine [3] allows to exept pratially a formation of the above-desribed supporting onial zones with the stressed state lose by hydrostatial ompression. In this ase the ylindrial ore both in the ylindrial and ubi speimen arouses the hoop tension stresses under ation of its axial ompression. In this situation a harater of a deforming and of a fraturing of the speimen is determined by its material properties, i.e. by its plastiity or brittleness. The brittleness degree is usually determined by a ratio of the ompression strength to the tension strength of the given material. Under ation of axial ompression the speimen from mild metal is flattened, the speimen from brittle metal (grey iron) is fratured along the diagonal surfae, formed by two 7

8 raks, developed along the both opposite trajetories of the prinipal stresses on its lateral surfae, and the speimen from stone (granite, ement) is fratured along the longitudinal raks. Considering the stressed state of a solid body under ation of axial ompression, one annot but mention an influene of the body form on its load-arrying ability. Nadai [5], on a base of the experimental researh results on axial ompression of several modifiations of the piped speimens from the burnt porelain, arried out by Westinghouse Laboratories (May,1941), offered a form of the piped speimen with its mean surfae in the kind of the hollowed hyperboloid of rotation with its thikness, inreased proportionally to its diameters. A question arises in onnetion with it: what one from multitude of the hollowed hyperboloids of rotation meets the requirement of the most strength in a ombination with the least weight? In his previous artile [1] the author showed that the trajetories of the prinipal stresses in the shell part of ylindrial speimen form two orthogonal families left- and right-handed of ylindrial helies, and on reahing by the speimen of its ultimate strength these helies are straightening oneself and form the nek in the kind of the hollowed orthogonal (right) hyperboloid of rotation. The inline angle of the straightened generatries of suh hyperboloid to its rotation axis is equaled to the inline angle of the helial trajetories of the prinipal stresses on the speimen external surfae and one makes up 45. At the same time, the examples of a test of the speimens from brittle materials on axial ompression, addued in the book [5], showed that Lueders lines, as refletion of trajetories of the prinipal stresses, are inlined on the greater then 45 angle. In fig. 31 (photo), just mention book, the inline angle of Lueders lines on external surfae of the ylindrial speimen from marble makes up 6. If one take that a ratio of the ompression strength to the tension strength is equal for marble and the burnt porelain, then it should be took 9 6 = 3 as the inline angle of the straight generatries of the hyperboloid mean surfae to its rotation axis. The greater thikness of suh hyperboloid body is to have been in its least ross-setion (nek) and one has to meet the equality ondition of the wall ross-setion area to the orifie ross-setion area in the hyperboloid nek. This quantity of the wall ross-setion area should be taken for the rest ross-setions of the hyperboloid, therefore its wall thikness should be dereased proportionally to an inreasing of its diameter. As to the hoop stresses, mentioned by Nadai, the hollowed hyperboloid of rotation not ontains suh omponent of its stressed state on a definition of this figure. The author explained this feature in his previous artile [1]. Completing a onsideration of the solid body state under ation of axial ompression, the author supposes to be expedient a onsidering of the question, onneted with the high-speed and superhigh-speed ation of ompressive fores on the elasti-plasti body, apable to a flow. The simple example of the quite high-speed ation of ompressive fore onto a body an be presented in the kind of the blows with usual benh hammer onto a flat butt-end of ylindrial bar from mild metal (aluminum, opper, steel). Under suh ation, when the effetive ontat duration of the hammer with the bar is smaller than duration of a running of the ompression longitudinal soni wave in the shortest from these two bodies in the diret and reverse diretions, the suffiient part of the hammer kineti energy is transformed into the loal plasti deformation of the bar. Ation of suh many blows leads to a spilling of the bar butt-end up to an origin of radial raks, developed from external ontour of this part of the bar. A thikness of the plasti deformation zone in the bar along its axis an be determined by expression 8 δ pl = l t h i t swh, where l h is a length of the shortest body, partiipated in the blow interation, t i is duration of the blow impulse, t swh is duration of a running of the diret and reverse longitudinal soni wave in the shortest body; in the given ase, for instant, the hammer is aepted as the shortest body. A thikness of the loal plasti deformation zone not exeed usually a half the bar diameter, and a ratio of the axial ompression stresses on the ends of this zone orresponds to a ratio of aousti stiffness of the bodies, partiipating in the blow interation

9 9 = ρ A ( ρ A ), σ σ δ h h h b b b Where σ and σ δ are axial ompressive stress at the bar butt-end and on a distane δ pl from it, respetively, ρ h, h, Ah and ρ b, b, Ab are density, the soni wave veloity and the ross-setion area of the hammer and the bar, respetively. A produt ρ is aepted to be alled by the aousti stiffness of the given material or its impedane, and a produt ρ A is offered [1] to be alled by the body blow stiffness. If in the onsidered ase one will suppose that the hammer and bar impedanes are lose by or equal eah other then previous equality takes the kind σ σ δ = A h Ab Let us suppose that the hammer ross-setion area is suffiiently larger then that of the bar. Than the axial ompressive stress at the bar butt-end will be larger then that on a distane δ from its butt-end. Therefore the spilling of the bar under the hammer blows is going on just at the bar buttend. Under the hammer every blow the bar ore part, experiening the longitudinal ompressive impulse, exerts the impulsive radial pressure onto the bar shell part. Under ation of the ore radial pressure in ombination with own axial impulsive pressure the bar shell part is spilt and one forms the irular ollar, inreasing its diameter with the hammer every blow. By a suffiiently quantity of the hammer blows, the radial spilling of the bar ollar is aompanied by origin in it of radial raks, loalized more or less uniformly along the ollar irular ontour. The addued example of the hammer and bar interation an be ompared with the following example. A water drop falls freely from height up to one meter onto the flat solid surfae. Supposing that the water drop as any spherial body ontains in itself the ore and shell parts with the ore radius ~ 79.4 % and a thikness of its shell part ~ 2.6 % of the drop external radius, as it is showed in the given artile Solution setion, the transient proess of the drop interation with the solid obstale an be presented in the following sequene. In a beginning of the drop ontat with the obstale, the drop spherial shell is deformed against the flat obstale. At the same time the spherial ore, beause of its braking, squashes under itself the drop shell part, arousing a transversal radial spreading of this part of the drop shell. The drop ore ontat with the flat obstale is also aompanied by its flattening and by its transversal radial spreading. The drop shell part, plaed over the ore part and round it, flows round the drop deformed ore and begins its transversal radial spreading under ation of its own braking and of a transversal radial pressing of the drop ore onto it. The desribed period is the period of transformation of the longitudinal momentum into its transversal radial momentum, aompanied by a formation of the irular water wave on the flat solid surfae. In the next period this water wave under ation of its hoop ompression begins its radial motion from away the drop fall entre and to the drop fall entre. The wave part, moving to the entre, forms the thin vertial fountain. The wave other part spreads, forming the ompat irular front ontour. Suh motion is aompanied by the resistane fores in the kind of the water internal frition and the water surfae tension. The hoop ompression of the spreading wave is dereased as its radius is inreased, and one this ompression is dereased up to naught and further one transforms into the hoop tension. Then the ompat irular front ontour of the spreading wave disintegrates to the series of the same separate jets. Diameter of these jets is dereased gradually under ation of the water surfae tension, and its stop is aompanied by elasti reoil with a throwing out the miro drops out of an end of every jets. H.E. Edgerton (1934) had made the well-known wonderful photo of the disintegration final stage of the milk falling drop onto a plate. Comparison of the desribed proesses of the blow ation on the bar from metal and on the water drop testifies to that a motion of mehanial energy in a solid body and fluid and also its aumulation and distribution in these bodies and fluids, orrespondingly its form, and transmission of mehanial energy in the ontat interation of the bodies and fluids between themselves all these proesses have the same dynamial wave nature. Differene in a behavior of suh material.

10 as metal from fluid under ation of external fores is stipulated by a presene of stati Young s modulus and the very high visosity in the solid body in ontrast to suh fluids as air, water. This thesis, to a ertain degree, is orroborated by H. Kolsly and D. Rader experiments [11] with the bars from polymethilmetharilat (PMMA) and polystyrene and also with water, put to an ation of impulsive load. M.A. Lavrentjev and B.V. Shabat [12] addued photos of longitudinal setion of the welded joints all over its ontat surfae of two parts in the kind of a oni bar and adjaent to it a shell and also of two sheets from mild metal, are made by means of the explosive, overing the external surfae of one of the joined parts. The vortex-wave struture of these welded joints orresponds exatly to the vortex-wave struture, arising in relative motion of two water (or air) stream, ontating eah other along initial flat surfae. The evolution sheme of suh vortex-wave struture development is presented in fig. 7 in author s artile [13]. The same struture, aompanying the irular interation of the air streams, is addued by M.D. Van Dyke in fig. 81 (photo) in his An Album [14]. The super-high-speed ation of the shaped harge explosion ompresses its onial metalli shell and transforms it into the high-speed jet from the unmelted metal [12]. Suh metalli jet goes through metalli barrier by means of its washing away without its melting. Now it is well-known the tehnology of a utting by the liquid (water) jet under high pressure (up to 5, atmospheres) of the different solid bodies, inluding metals. The physial adequay will not be broke by omparison of the water avitation in the onvergeddiverged nozzle (Venturi nozzle) with frature of the ore part of ylindrial speimen from mild steel under its axial tension, in author s artile [1]. In both these ases material water, steel is in its state lose by hydrostati tension. Let us ompare the water stream state in the pipe with the stressed state of ylindrial bar from steel under ation of axial fore. The water steady motion in the pipe is going on under ation onto it of the invariable fore of a pressure drop and one is determined by the water internal frition visosity against the pipe wall. The internal frition as a dynamial fator determines, in one s turn, the stati pressure distribution along and aross the stream, ontaining the axial and two transversal radial and hoop stati omponents of the stream total head vetor (STHV) [15]. Geometrial sum of these stati omponents with the dynamial omponent in the kind of the stream veloity head determines ompletely the stream energy state in any ross-setion of the stream. In this situation one annot but pay attention to a dynamial harater of the stream in the kind of its motion and in the same time to the pressure stati harater along and aross of the same stream. I.e. the stream moves, but pressure in it is stopping. In ontrast to it, ylindrial bar (speimen) and, orrespondingly, its stressed state remain immovable at its testing on axial tension or ompression, at least on stage of its elasti deformation. Suh differene in behavior of the water stream in the pipe and metalli speimen is onneted with a lak of Young s modulus in water, owing to that water possesses always additional degree of freedom in the kind of its infrastrutural mobility. Therefore the water stream in pipe an be moved and in the same time its stressed state an be immovable. This stati stressed state in the water stream is quite equivalently to the stressed state of metalli bar under ation of longitudinal fore, stipulated by its own weight. The omponents of the total head vetor of the water stream in pipe form in this stream, under ation of its frition against the pipe wall, the same two families of trajetories of the stati pressure ation in the kind of ylindrial helies as the trajetories of the prinipal stresses in the steel ylindrial bar under its axial ompression by its own weight fores. A fitness of the mahine elements to transmission of the torsion moment is determined by the test results of mainly ylindrial speimens by means of the torsion-testing mahines. In his book [5] Nadai dediated to this kind of a testing the separate hapter, in whih he addued two methods of a alulating of the shear (tangential) stresses in ylindrial speimen under ation of the torsion moment. Side by side with it Nadai addued some results of experiments, arried out, in partiular.. by J.H. Pointing (199, 1912), E.A. Davis (1937) and H.W. Swift (1947). These results, in ontrast to the theoretial finding, testify to the fat that the speimens from the piano wire and in the kind of ylindrial bar from mild opper, brass and aluminum lengthen oneself under ation of the torsion moment. Swift showed also that a hange of the moment diretion is aompanied, in the 1

11 beginning, by its shortening aording to a dereasing of the torsion moment of initial diretion and than by the new lengthening aording to an inreasing of the reverse diretion moment. Davis deteted experimentally an origin of the old-work strengthening in the ore of ylindrial speimen from mild opper in the result of its plasti twist. He asertained that the hardness rises as an approahing to the speimen ross-setion entre. On these grounds Nadai surmised a possibility of existene of normal (prinipal) axial, radial and hoop stresses in ylindrial speimen under ation of the torsion moment, but he did not develop this idea. This idea is not developed till now neither in the theory of elastiity nor in the theory of plastiity, although solution of a problem on the shear stresses in this kind of a loading an be dated, aording to Nadai [5], by To eluidate a question on possibility of existene and of a distribution harater of the prinipal stresses in ylindrial speimen from statistially isotropi material under ation of the torsion moment, it should be took into aount the well-known equivalene of the stressed state in a pure shear and in ombination of normal (prinipal) stresses. This equivalene allows presenting the stressed state of ylindrial speimen under ation of torsion moment both in the kind of a pure shear and in the kind of ombination of the prinipal stresses as substituting mutually eah other. An appliation in speifi ase of either of approah to the speimen stressed state analysis is determined by the speimen material properties. In partiular, a taking into aount of normal stresses, determining the kind of the ylindrial speimen frature under ation of the torsion moment, is to be neessary for the brittle material, having its tension strength in 2 5 times less than its ompression strength. In this ase it is neessary to onsider the ylindrial speimen stressed state by means of the prinipal tensioning stresses, ating along the ylindrial helies, having its angle equal 45 and its diretion aording to the diretion of the torsion moment ation. When the tension stress along this one family of the helies reahes the tension strength of the brittle material, then suh speimen is fraturing on the helial surfae with its external helial edge, oriented orthogonally to the tension stress trajetory. Although the speimen shell part, underwent to an ation of the hoop tension stress, exert radial pressure onto the speimen ore part, a lengthening of suh speimen is insignifiant beause of the low tension strength of brittle material. To eluidate the stress state harater and the frature shape of ylindrial speimen from suffiiently plasti material under ation of the torsion moment, the given artile author has arried out several experiments in the kind of a twisting by hand of the wire piees from mild aluminum (of type 2L4 British Standards Speifiations) with its diameter 1.8 mm. These experiments showed that a frature of suh speimens was going on after that as the inline angle of helial lines (Lueders lines) reahed ~ 2 on the wire surfae. I.e., in the given ase the hoop tension stresses in the speimen shell part and, orrespondingly, its radial pressure onto the speimen ore part are found to be in ~ 2.5 times greater than under 45 angle. In that way the speimen shell part by its thikness ~ 16 % of the speimen radius is under ation of the hoop and axial tension stresses in its orrelation, determined by a tangent of ~ 2 angle of the helial trajetories of the prinipal stresses and also of internal pressure, stipulated by the ore part stiffness. In one s turn, the speimen ore part is in lateral ompression state under ation of onto it of the shell part and also in the shear state under ation of its twisting. In this situation one half the torsion moment energy, applied to the speimen, is realized in the kind of the shell part stressed state and the energy other half is realized in the kind of the ore part stressed state in aordane with equality of the polar moments of inertia of the speimen ore and shell parts. The tested speimen frature begins always by the partial rupture of one of the helial oils in the speimen shell part and then one is aompanied by the speimen transversal utting by means of mutual turn of the speimen butt-ends, adjoined eah other in the ut flatness, around the rest intat part of the partially ruptured helial oil. In that way the speimen shell part with its relative thikness ~ 16 % and its ross-setion relative area ~ 29.3 % is strethed as a ylindrial spring under ation of the torsion moment and one squeezes the speimen ore part with its ross-setion area ~ 7.7 %. In the result the speimen ore part is lengthened in the beginning elastially, then plastially and one arries along together with oneself the speimen shell part, arising thereby the speimen lengthening. In the ase of the speimen from mild opper these onditions will arise the 11

12 old-work hardening of the speimen ore part, deteted by Davis in his above-mentioned experiment. The desribed lengthening of the speimen promotes to an inreasing of the helix angle in its shell part. The above-mentioned initial partial rupture of one of the helix oils is going on just in a moment, when a dereasing of the helix angle under ation of the torsion moment is balaned by an inreasing of this angle beause of the speimen lengthening. Strong influene of the speimen lengthening onto an inreasing of the helix angle and, orrespondingly, of the shell axial tension stress is stipulated by that the ore part ross-setion area is more in 7.7 / 29.3 = 2.4 times than the shell part ross-setion area. In the above-desribed experiments of the given artile author the aluminum wire frature under ation of the torsion moment was not aompanied by the visible residual hange of the speimen form or by its visible residual loal deformations as it takes plae at the ylindrial speimen tension or ompression test. The only shear harater of suh frature and small quantity of the helix oil angle testify to a high dutility of the ommerial pure aluminum. In this onnetion the author supposes advisable to reommend to a plastiity valuation of any material by means of the helix angle (Lueders line angle) at the ylindrial speimen test by the torsion moment. The approximate orrespondenes are the followings: 2 < ϕ 2 - high-dutile material (the ommerial pure aluminum); ϕ 3 - dutile material; 3 4 ϕ - low-dutile material; ϕ = 45 - brittle material; ϕ - high-brittle and/or anisotropi material (timber). The asertain of existene and the eluidation of the prinipal normal stresses and its orthogonal omponents in ylindrial speimen under ation of the torsion moment allows also to disover nature of the stress onentration under ation of the torsion moment in zone of the speimen diameter sharp hange. A quantity of the prinipal stresses and, orrespondingly, of its omponents in the speimen part of the lesser diameter is found to be larger then that is in the speimen part of the grater diameter proportionally to a orrelation of its polar (torsion) moments of resistane, i.e. to ( D ) 3 1 D 2. In the other words, even at the not great differene in diameter of these parts of the speimen the quantities of the prinipal stresses and, orrespondingly, its omponents are found, in the speimen lesser diameter part, to be so larger that the stressed state of the speimen greater diameter part an be left out of aount. Therefore below it is onsidered the stressed state only in the speimen part of the lesser diameter, ontained, in one s turn, its ore and shell parts. The transversal omponent of the prinipal stresses in the kind of the hoop tension stress, ating in the shell part, squeezes the ore part and provokes its lengthening. In the result the shell part is loaded by axial tension stress in addition to the axial tension stress, as a omponent of the prinipal stresses. Beause of its lengthening, the ore part exerts axial pressure onto the speimen greater diameter part and thereby one provokes the loal additional tension stress in the shell part. Although the axial ompression fore in the ore part is balaned by the axial tension fore in the shell part, the loal axial tension stress in the latter is more in 2.4 times than the loal axial ompression stressing the ore part. In that way the shell part is loaded by a sum of the following three omponents of the axial tension stress: as the prinipal stress omponent, as the omponent, stipulated by the ore part lengthening, and as the omponent, stipulated by a pressing of the ore part onto the speimen greater diameter part. The first two from these omponents are distributed uniformly along the speimen axis; the third omponent has the loal harater. Ratio of a sum of these three omponents to a sum of the first two omponents is the so-alled stress onentration fator, stipulated by the torsion moment ation, applied to ylindrial speimen, having a sharp hange of its diameter. In that way the stress onentration is retribution for non-observane of the smooth onjugation of trajetories of the prinipal stresses. 12