To the Strength First Problem Full Solution: Mechanics of a Necking

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1 To he Srengh Fir Problem Full Soluion: Mechanic of a Necking S. L. Arenjev 1 Phyical-Technical Group Dobrolubova Sree,, 9, Pavlograd, Dneproperovk region, Ukraine Eenially new approach o analyi of inernal force, ariing in cylindrical rod under acion of an axial enion force, ha allowed o deec he hree-dimenional axiymmeric re ae. According o he new concepual model he axial enion force caue he angenial hoop and radial ree ide by ide wih axial (normal) re in he rod volume. A compleion of Lame oluion of he problem on he re ae in he hickwalled cylinder ha allowed o acerain a muual direc and revere connecion of he angenial and radial ree wih he axial re alo in he rod under acion of an axial enion force. The new approach for he fir ime ha allowed o give he full phyically adequae and mahemaically ufficienly ric decripion of a change of he iniial cylindrical form of a mild eel rod under acion of an axial enion force on all age of i deforming, including he necking, he fracure proce and a view of he fracure urface. The new approach ha allowed on he unied mehodological bae o elucidae alo a number of he queion, bound wih an axiymmeric form of he oap oluion film beween wo ring, wih he breaking up of a liquid free je ino drop, wih he caue of a buckling of he long ube under acion of inernal preure, creaed in i by he reed and moved fluid in i and oher. PACS: b; b; 46.5.Cc; x; ; a; p; 6.0.Mk; n; Jk; Bx; B; Bc; La Inroducion One of he mehod of experimenal deerminaion of phyical and mechanical properie of rucural meal and in he fir inance of eel i he enion e of a pecimen wih i working par in he kind of a cylindrical rod. Dimenion of hee pecimen are andardied, and i diameer i uually 10-0 mm and he relaive lengh of i working par i L/D = The pecial enion-eing machine are ued a he echnical and reearch iniue and a he machinebuilding plan for a deermining of phyical and mechanical properie of he new rucural meal and alo for he conrol of he rengh and plaiciy level of he machine, apparau, rucure par. The enion e for a deermining of aic rengh are conduced under he limied velociy of he pecimen enion. Side by ide wih rengh he plaiciy i one of baic characeriic of eel, deermining i applicaion. Relaive decreae of he pecimen cro-ecion area a a fracure during he enion-e i i plaiciy meaure. The relaive narrowing for mo of he rucural meal i uually 50 ± 0 % and one i conneced wih a decreae of he pecimen working par diameer in he whole and wih a forming of a o called neck. Cup and cone i a name of he fracure urface in he neck one, on he profeional lang. Taking ino accoun he grea imporance of he plaiciy for rucural meal and a lack of he aifacory explanaion of a change of he pecimen form during he enion e, he auhor of he given aricle e himelf a an objec he creaing of raional concepion, which decribe he behavior of he eel pecimen in he kind of cylindrical rod under acion of a enion force phyically adequaely and mahemaically ufficienly ricly. One of baic requiremen o he concepion i, on he auhor opinion, a uniy of approach o decripion of a change of he pecimen form on all age of i deforming from a beginning of he enion e up o he pecimen fracure. 1 Phone: ( ) (home, Ru) up777@ukr.ne; pglecu@yahoo.com

2 Approach The approach bai o a oluion of he raied problem coni in he following wordy formula: - a longiudinal enion of a cylindrical rod of ioropic maerial caue in i he hreedimenional axiymmeric field of inernal force, which are he angen o an orhogonal family of he cylindrical dexrore and inirore helice; hee oppoie helice are he acion rajecorie of he inernal force in he axiymmeric rod under acion of a enion force; in he elecromagneim heory he uch ype line i called he field line ince Sir Faraday and in he coure of Srengh of Maerial one are called by he principal re rajecory; - on he rod urface hee oppoie helice form a yem of he longiudinal enion force and of he ranveral hoop enion force of equal o hem in a quaniy; - he wall boundary layer of he rod ha he cro-ecion area, equal o a half of he rod croecion area and hi layer i under acion of he ranveral hoop enion force, increaing o he inernal boundary of he layer; - hi layer, in he kind of he hick-walled cylindrical hell, compree he core rod; - imulaneouly boh he rod core and compreing of i he rod hell i under acion of he ame longiudinal enion force, equal in um o he enion force, applied o he rod. Soluion In accordance wih he approach wordy formula o a oluion of he given problem a preenaion of he rod a a volume whole in he kind of he rod hell and he rod core allow o ue G. Lame formula (1833) for an analyi of he ranveral hoop and radial ree and of hem ineracion preure. Fig.1 preen diagram of he force and ree for cylindrical rod under acion of he axial enion force. Beforehand i i neceary o ake noice, ha he hoop ree are deermined by a um of wo ranveral force accordingly o wo componen dexrore and inirore of he orhogonal family of a field of he cylindrical helice d d + = ; =. (1) Every of hee wo oppoie cylindrical helice ipulae an acion of he orque in he enioned rod in a correponding direcion: clockwie and ani-clockwie. Thee wo orque equal in a quaniy and acing in oppoie direcion no deroy each oher, bu one creae he hoop enion re in he enioned rod. Side by ide wih i an orhogonaliy of he field line deermine alo an equaliy of he hoop and axial ree on he rod urface =. () Under acion of he hoop ree he rod hell render a preure on i core. A he ame ime he rod hell, in conequence of he grea quaniy of a raio of i cro-ecion radii r = r 0.707, 1 r can be aribued o he hick-walled cylinder hell under acion of inernal preure. In he conidered cae Lame formula for a deermining of he hoop re on exernal urface of he rod hell ha he kind r r =, (3) 1 ( r r ) = p (1 + ) = p = r r1 r i.e. he preure quaniy on a boundary of he rod hell and he rod core i equal o a quaniy of he hoop re on he rod exernal urface and imulaneouly one i equal o he longiudinal (axial) re.

3 The hoop re in he rod hell on a boundary wih he rod core i, accordingly, r1 r ( r = r ) = p (1 + ) = 3p or 1 ( r= r ) = (4) r r1 r1 A ha he radial ree in he rod hell are changed wihin he limi from a ero on he exernal urface up o = r p on i inernal urface. Simulaneouly a well-known condiion on a conancy of algebraic um of he hoop and radial ree in he hick-walled cylinder under acion of he inernal preure + ( r ) = con i aified all over he cro-ecion urface of he rod hell and one confirm a conancy of longiudinal re on hi urface. The hoop and radial ree in he rod core, aking ino conideraion i compacne, ha a imple kind = r = p = all over i cro-ecion urface. The adduced expreion for a deermining of he ree in he cylindrical rod under acion of he enion force are ju boh on he elaic and plaic deformaion age while he rod reain a cylindrical form. A he ame ime he reul of he numerou enion e of pecimen in he kind of he cylindrical rod of he plaic meal and, in paricular, of a mild eel how ha a form of uch pecimen i eenially changed during a proce of i plaic deforming, alhough one remain axiymmeric. The general feaure of he form change of uch pecimen a he plaic deforming under acion of he enion force are diagrammaically preened in he aricle [1]. The mo ypical form in he proce of he pecimen irreverible lenghening i he following: - he local narrowing of cylindrical par near one of he pecimen end; - a diameer of he pecimen cylindrical par i equal o he local narrowing diameer, menioned on previou age; he pecimen cylindrical par i dilaed moohly up o he iniial diameer near boh end of he pecimen; - he pecimen cylindrical par ake he hape of a core; - he local neck arie in he narrowing par of he core. Geomerical analyi of he lied form allow o acerain ha every of hem can be conruced by an adding only wo urface of roaion o he iniial cylindrical urface: he one-hollowed hyperboloid and an evolvenoid. The choice of he one-hollowed hyperboloid i condiioned by wo baic reaon. Firly, hi urface can be conidered a he cylindrical urface modificaion if a generarix of he laer will be pli on wo croed inclined raigh line, which one are ymmerically inclined relaively of he roaion axi and one connec by ielf he circumference in he cylinder bae. The very viual decripion of a mehod of ranformaion of a cylindrical urface ino he one-hollowed hyperboloid urface i adduced by prof. D. Pedoe in hi beauiful book []. According o hi mehod a cylinder can be conruced by mean of wo ame wire ring, conneced each oher by he parallel hread. Then a urn of one of he ring in i plane on ome angle, under a condiion of he ame enion of all hread, ranform he cylinder ino he one-hollowed hyperboloid of roaion. In conra o he cylinder of roaion, poeing only one family of he generarice, a urface of he hyperboloid conain wo familie of generarice. The econd family of he generarice can be formed by a urn of ha ring from he iniial poiion in oppoie direcion ha angle. Naurally, a heigh of uch hyperboloid will be le of a heigh of he iniial cylinder proporionally o he incline angle of i generarice. Secondly, he muliude of he one-hollowed hyperboloid conain a modificaion, in which he croing angle of i generarice, equal 90 0, and imulaneouly i inerecion angle wih he roaion axi, equal A heigh of uch hyperboloid i equal o he chord lengh of he circle quarer in i bae. Thu, a heigh of uch hyperboloid i rericed in conra o a cylinder. The oher mehod of he uch hyperboloid conrucion i ha a cube i aken a an iniial geomerical figure. If he upper and lower ide of hi cube will be circumcribed by 3

4 circle and he diagonal of one of he cube laeral ide will be roaed abou he verical axi, paing hrough he cener of he circle, he formed urface of a roaion will be preen by ielf he one-hollowed hyperboloid wih he heigh, equal o i neck diameer. One of he feaure of uch hyperboloid i ha he cro-ecion area of i neck i equal o a half of i bai area, which approximaely correpond o he relaive narrowing of pecimen of he a mo of he rucural meal under a enion e. Thi pecial modificaion of he hyperboloid i called by he regular one-hollowed hyperboloid. In he conex of a given problem hi hyperboloid modificaion can be funcionally called he orhogonal one-hollowed hyperboloid of roaion. Conformably o a enion of he cylindrical pecimen by he axial force, an influence of i form on i re ae i in he following. While he pecimen reain he cylindrical form, he orhogonal family of he field line caue he hoop ree in i hell, which compre i core by preure, equal in a quaniy o he enion re. Such acion on he pecimen core promoe preervaion of i coninuiy during he plaic flow age up o a reaching of he enion force culminaion (ulimae rengh). Beginning from hi momen, energy of he field of he enion force reache a level of he inrarucural energy of he pecimen maerial, preervaing i iniial form, and hen one exceed he laer. Now he orhogonal family of he inernal force pli a generarix of he cylindrical urface on wo croing inclined raigh line ha are ymmerically inereced wih he pecimen axi and, in ha way, one ranform he pecimen cylindrical form ino a hyperboloid form, called by a core. According o experimenal daa [1] uch core i formed by a urn of one of he bae circle of iniial cylinder in a beginning on an angle approximaely ± 3 0 and hen up o approximaely ± The hyperboloidal core i imulaneouly formed on he whole lengh of he pecimen working par and becaue of i he lenghening velociy of he pecimen become commenurable wih he acion velociy of he enion-eing machine. In a reul he enion force, recorded by he machine, i decreaed. A decreaing of he pecimen cro-ecion area in he core neck one lead o a ubequen increae of he pecimen re ae and reul in he local narrowing in he kind of he orhogonal hyperboloid. Now, a la, he above menioned cylindrical helice raighen oneelf and coincide wih he orhogonal hyperboloid generarice. In he Srengh of Maerial coure uch local narrowing i called by a neck. The neck i formed quickly and accompanied by a decreae of he pecimen cro-ecion area on ~ 50%. Accordingly o uch ranien and grea ranveral deformaion a lenghening of he pecimen i going on again wih a velociy commenurable wih he acion velociy of he enion-eing machine. The machine again record a decreae of he enion force. In he necking proce he mooh conjugaion of he orhogonal hyperboloid urface wih he hyperboloidal core urface i formed by he roaion urface wih i generarix poeing a variable curvaure. The mo accepable curve from he piral family for uch generarix i a circumference evolven. While he enioned pecimen reain he cylindrical form he developmen angle of hi evolven i equal o a ero. In he momen of a forming of he neck, before he pecimen fracure, he developmen angle of hi evolven i ~0 0 in a poin of i conjugaion wih he urface profile of he orhogonal hyperboloid. Fig. preen diagram of a conrucing of he longiudinal profile of he enioned cylindrical rod on he necking age under acion of he axial enion force only. Fig. 3 how ha under acion only of he laeral hydroaic preure he developmen angle of he evolven reache he maximum quaniy `. The above menioned raighening of he cylindrical helice and i coincidence wih he orhogonal hyperboloid generarice reul in a ceaing of an acing of he hoop ree in hi par of he pecimen and correpondingly in a ceaing of a preing of he pecimen core by he pecimen hell. A decreae of he pecimen cro-ecion area on ~50% in he neck correpond o a doubling of he axial enion re boh in he hell and core par of he pecimen in i minimum cro-ecion. Under hee condiion he pecimen hell poee a poibiliy for he ubequen plaic deformaion owing o a free exernal urface. In conra o i uch poibiliy of he pecimen core i rericed by a hoop rigidiy of he pecimen hell. The rericion of poibiliy of a narrowing of he pecimen core in he neck one under acion of he exremal axial enion re lead o a developing of he hree-dimenional enion of he pecimen maerial in hi one. Thi 4

5 re ae i in he beginning accompanied by a ome addiional narrowing of he neck, hen by appearance of a poroiy of he pecimen core in hi one and, a la, by a developing of a ranveral crack from he pecimen axi in he cro-ecion, correponding o he conjugaion poin of he urface profile of he orhogonal hyperboloid and he evolvenoid (poin C or poin E on fig. ). The ubequen fracure of he pecimen hell i going on plaically by he way of a hear owing o he deformaion freedom of i exernal urface and parially of i inernal urface. Uually uch form of he pecimen fracure i called by cup and cone. One of feaure of he onehollowed hyperboloid of roaion i ha he cro-ecion circular form of i neck i lighly ranformed ino an ellipe. Therefore omeime [3] he iniial ranveral crack in he pecimen core alo ake he ellipe form. The ubequen fracure of he pecimen hell i ared near he end of he ellipe major axi and one i compleed by he neck-and-pli fracure. In he frame of he above aed approach he mechanic of he local narrowing, preaded along he pecimen working par on he iniial age of i plaic deformaion, can be preened by he following way. In an iniial ae he polycryalline rucure of he mild eel i aiically ioropic and one conain, on he experimenal and heoreical reearch daa [4], he rigid inercryalline laice of a cemenie and he imperfecion of he cryal in hemelve in he kind of dilocaion [5-7]. On he iniial age of he pecimen plaic deformaion an energy of he above decribed axiymmeric volumeric field of he inernal force, excied by he axial enion force, i ufficienly only for he ubequen fracure of he inercryalline cemenie laice and only for a deforming of ome cryal, poeing a correponding orienaion of he dilocaion. A a reul of a preading of he local narrowing along he pecimen lengh i rucure i ranformed ino he aiically orhoropic. In he oher word, he iniial age of he pecimen plaic deformaion i a age of he volumeric rucural adapaion of he pecimen maerial o a rucure of he volumeric field of he inernal force, caued by an axial enion force. A urface profile of uch local narrowing i, a before, a combinaion of he hyperboloid wih he evolvenoid. The reul of a oluion of he problem, aed in he given aricle, eify o ha he pure uniaxial reed ae i poibly only for he rod wih i diameer, equal a ero. In he hree-dimenional body he field of he inernal force i alo he hree-dimenional. A he ame ime a poibiliy of a reoluion of a hree-dimenional field of he inernal force on he orhogonal componen i kep in full. Dicuion of reul Comparion of he approach and oluion of he aed problem wih he reul of he well-known experimenal reearche eifie o he following. Lueder [8] line eify o he acion poibiliy of he orhogonal family of he field line in he kind of he dexrore and inirore helice in he cylindrical pecimen of he mild eel under acion of an axial enion force. E. Bollenrah, V. Hauk and E. Owald [9], uing he roengenography mehod, had inveigaed he reidual ree in he mild eel cylindrical pecimen afer i preliminary enion o 11%. Thi reearch ha allowed o acerain ha an area, occupied by he reidual axial compreive ree, ha a form of a ring along he exernal conour of he pecimen cro-ecion and hi area i equal o 50% of he pecimen cro-ecion area. The ame area i occupied by he reidual axial enion ree in he pecimen core. S. P. Timohenko [10] cie phoo of he hor (L/D = 5.4) hin-walled (δ /D = 0.07) ube, enioned up o a fracure. Thee experimen were conduced by E. A. Davi (Weinghoue Reearch Laboraorie) [11] and one how ha a change of a form of he ubular pecimen under acion only of a enion force correpond ufficienly o i in he cae of he olid cylindrical pecimen. Among he grea volume of informaion, dedicaed o he experimenal reearche of he mechanical properie of meal by mean of he olid cylindrical pecimen, P. W. Bridgman [1] reearche are brighly diinguihed by a combinaion of he enion force wih he laeral hydroaic preure. Thee reearche how ha an acion of he laeral hydroaic preure ide by ide wih 5

6 he enion force lead o a decreae of he cro-ecion area of he pecimen core in he neck one. Under ufficienly high laeral preure he pecimen core diappear in hi one and he plaic fracure i going on in he pecimen hell. Subequen increae of he laeral preure lead o a narrowing of he pecimen ino a poin in he neck one. V. I. Feodojev [13] inform ha a plaic deformaion and fracure of he cylindrical rod under acion of he ufficienly high laeral hydroaic preure in he abence of he axial enion force ha he characer and he hape quie imilar o a behavior of ha rod under acion of he axial enion force only. Uing a fig.1 f, g, i i eay o ee ha an acion of he laeral hydroaic preure allow no only o compenae he enion force beween he core and he hell of he rod on he necking age, bu alo o enure a compreing of he rod core by i hell. A ha he ubequen enion of he rod will be accompanied, in he beginning, by a narrowing of he rod core and hen by a narrowing of he rod hell ino a poin, imilar o he above menioned Bridgman experimen. Thu, he reul of he experimenal reearche corroborae a correcne of he hee of he approach wordy formula o he given problem oluion and one allow o affirm an exience of he direc and revere connecion of he longiudinal (axial) and ranveral (hoop and radial) ree owing o an exience of he orhogonal family of he field line (he principal re line) a a bae of he field of inernal force in cylindrical rod. Thi bae exi virually in he abence of exernal influence; hi bae exi really on all age of elaic and plaic enion. Thi bae exi in he cylindrical rod alo under acion of he laeral hydroaic preure only. In conra o he explanaion [14], in which he necking i bound wih imperfecion of he rod iniial cylindrical form, he oluion, given in hi aricle, uppoe a geomerically ideal cylindrical form of he rod. A regard he direc and revere connecion of he ranveral hoop and radial ree wih he axial re, he above menioned reul of Davi experimen [11] wih he hor hin-walled ube of he mild eel under acion of a combinaion of he axial enion force and he inernal hydroaic preure eify o he following: he longiudinal reidual deformaion of he ubular pecimen i correpondingly decreaed by 3; 39 and 43%, when he deigned hoop ree under acion of he inernal preure were ome leer, equal and greaer of he deigned axial re. If in Bridgman experimen he exernal (laeral) preure wa a caue of he axial enion of he olid cylindrical rod up o i earing, in Davi experimen he inernal preure ha became an obacle in he lenghening of he ubular pecimen under acion of he enion force. Naurally, he eel ube canno be ufficienly horened under acion only of he inernal preure becaue of he relaively mall quaniy of Poion raio of eel. In hi cae he rubber ube i he be example owing o a quaniy of Poion raio of a rubber equal approximaely o 0.5. Thu, in he frame of he approach, aed in he given aricle, wih he aking ino accoun of he cied experimenal reul he above menioned well-known condiion of a conancy of he axial re in a cro-ecion of he hick-walled cylinder under acion of he hydroaic preure hould be decribed o: in he cae of he inernal preure acion + ( r ) =, (5) in he cae of he exernal preure acion = ( r ) =. (6) A applied o he enioned cylindrical rod he expreion (5, 6) are ame correc for boh i hell and core par and in a whole. The expreion (5, 6) complee a oluion of Lame problem and one permi o olve he dragged ou paradoxal iuaion, bound wih a buckling of he long ube under acion of he inernal hydroaic preure, creaed by pion or by he aic head of a fluid, flowing hrough he ube. J. G. Panovko and I. I. Gubanova [14] declare ha uch ube loe he aic longiudinal abiliy, no experiencing of an axial compreion force in general. Feodojev [13] wrie on a fale noion ha in he abiliy queion by Euler he inernal compreion force play he baic role and he add acually i i no o. A he ame ime hee auhor adduce he righ oluion by Euler, applying formally he axial compreion force. I i 6

7 clear, ha uch aemen are baed on he numerou experimenal reul bu no he clear heoreical noion of a given phenomenon. Acerainmen of a connecion of he ranveral force wih he longiudinal force and a oluion of he problem on he inernal force, acing in he cylindrical rod under acion of an axial enion force, adduced in he given aricle, clearly preen a mechanic of an origin of he axial inernal compreion force in he ube under acion of he inernal preure boh under he aic condiion and a fluid flow hrough he ube. Ju he experimen, conidered by auhor [13, 14], are an evidence of he direc and revere connecion of he ranveral and longiudinal inernal force in hee cae. The modern approach ufficienly correpond o J. C. Maxwell concepion (1856): when (he forming energy) reached a cerain limi, hen he elemen would be fracured and alo o M. T. Huber (1904) and A. Foppl and L. Foppl (194) approach, cied by Timohenko in hi exbook [10]. Timohenko adduce he example of a preenaion of he rod re ae under acion of a enion force in he kind of a um of he hree-dimenional enion and he pure (imple) hear. The modern approach, aed in hi aricle, preen he re and rain ae field in i concree form wih aking ino accoun of he body form, of i maerial and he kind of he load, applied o i. The reul of a reearch, adduced in he given aricle, can be applied o a number of he ufficienly far phere from i. In paricular, he oap waer oluion film, enioned beween wo ring, accep ju he onehollowed hyperboloid form, enuring a raighforwardne of he urface enion force in he abence of a conrucive rigidiy of he oap oluion. Thoe, who bind a form of uch film wih a caenoid [15], fail o keep a principle: peak wha you know. A drawing of graviaion i miplaced in hi cae. From wo poible urface of a roaion cylindrical and hyperboloidal he oap oluion film, conrolled by he inermolecular ineracion energy, elec he laer on a principle of he energy minimum for a creaing of he form and correpondingly of a minimum of he urface area. Oher example, alo bound wih a liquid, i a breaking up of he free falling waer je, ouflowing ou of an orifice (ube), ino drop [15]. In hi cae an increae of he je fall velociy under acion of graviaion caue he longiudinal enion force in he je urface and, a a conequence, an increae of he ranveral enion force. In reul of i he je i narrowed imilar o he core of he enioned eel rod. The mooh narrowing of he je increae ponaneouly he ranveral enion force, leading o he necking in he kind of he one-hollowed hyperboloid. Then a combined acion of he longiudinal and ranveral enion force lead o a narrowing of he neck in a poin and o a breaking up of he je. The breaking up i going on very quickly and one i accompanied by a harp rebuilding of he hyperboloid form ino wo emiphere. Thi ranien proce, in one urn, i accompanied by a rong wave impule, acing ino he je upper and lower par. Under acion of hi impule he longiudinal and ranveral force of he urface enion ear almo inanly he je lower par ino drop. Thu he decribed very upper break of he waer je i a generaor of i breaking up ino drop. Nex example, alo bound wih he hyperboloid form, i form of a bell from he grea a he church belfry o he hip bell, hand bell. Sir Rayleigh wrie in hi fundamenal work [16] a regard a form, acceped for he church bell, one i no received he aifacory explanaion ill now. The Grea Sovie Encyclopedia [17] conain he following houghful decripion: he bell ha a form of he hollow pear wihou i lower par. Prof. Pedoe noice in above menioned hi book [] ha he neck cro-ecion of he one-hollowed hyperboloid, conruced by mean of wo wire ring and he hread beween i, ha he ellipe form. The auhor of he given aricle ha availed oneelf of a diveriy of he modern form of he polyehylene bole and ha purchaed a bole, conained he core wih 7.6 cm neck diameer, and he 9 cm diameer of i boom and a diance beween he boom equal 1 cm. Boh boom of hi bole have he kind imilar o a phere and one have a ufficien rigidiy of i form. A he ame ime hoop rigidiy of he core urned ou by exremely low. The lighe drawing ou of air ou of he bole ranform righ away a round cro-ecion of he core neck ino ellipe. The lighe upercharging of he bole ranform he 7

8 ellipe ino a circle. The core bae (he bole boom) in boh hee cae keep he round form of i cro-ecion. A apping by a finger ip along he core from one boom o he oher ha allowed o deec on boh ide from he core neck a ingly bel, where he apping, no caue he characeriic ounding. The apping ouide hee ilence bel in boh he core neck one and he core bae caue he characeriic low-frequency ounding wih i aenuaion during approximaely 7 econd. Thee ilence bel i locaed in he juncion of he hyperboloid neck concave profile wih he convex profile beyond he neck. The ame apping on a urface of a cylindrical polyehylene bole no caue he pure one and la no more 1.5 econd. In a nex experimen wih he core bole he apping wa accompanied by a upercharging of he bole. The experimen ha hown he rong direc connecion of he ound frequency wih he upercharge preure a he ame duraion of he ound aenuaion. Then he core bole wa hu off by he hreaded lid and placed in he warm air ream. Under hi condiion he apping ha allowed o acerain ha he heaing in limi from ~ 0 0 up o ~ 40 0 C lead o a riing of he ound one pracically on an ocave. An obervaion of he core urface and he liening of i in a proce of he ound ocillaion have allowed o acerain ha ju he core neck one i a bae ource of he ound radiaion. For all hi he core neck cro-ecion accep an ellipe form wih a changing of i major and minor axe a every cycle of he ocillaion and correpondingly o i he direcional diagram of he ound radiaion ha four peal, and one of hem coincide wih a place of he apping. So, he baic conrucive feaure of he bell can be preened in he kind of wo verion. Accordingly o he fir of hem hee feaure are he following: - a form of middle urface on he bell laeral hell i he one-hollowed hyperboloid par, conained he neck par and adjoining o i from below he diverging par; - an upper par of he bell laeral hell i rericed by a roof in he kind of a phere egmen wih he angen angle o i bae plane ~ 35 0 ; he roof conain in i cenre he bell hanger bracke; - a lower par of he bell laeral hell i broadened o a diameer equal of a heigh of he bell laeral hell and one ha along he lower edge a hickening he percuive bel; - in he whole he bell laeral hell ha approximaely he ame relaive hickne. In hi cae he bel hell no conain he ilence bel. Accordingly o he econd verion he lower par of he bell laeral hell preen by ielf he convexo-convex hell, forming he ilence bel wih he bell hyperboloidal neck. Fig. 4 how diagrammaically he menioned verion of he bell conrucion. Owing o he ilence bel, placed beween he bell neck and he bell lower par, hee wo par of he bell laeral hell ocillae in oppoie phae. In boh decribed verion frequency of he ound ocillaion, radiaed by he bell, i inverely proporional o i neck diameer. The decribed rucure of he bell form allow o give he concree recommendaion for an auring of he radiaed ound one, bu i overep he frame of he given aricle. Alhough a deecion of he ilence bel allow o explain why he iniial ranveral crack in he core of he enioned cylindrical pecimen, a aed above, arie ju in he plane, paing hrough he juncion of he hyperboloid and evolvenoid urface. The fac i ha he deformaion and plaic flow of he pecimen are accompanied by he longiudinal and ranveral ocillaion of a ufficienly high magniude. A change of a curvaure of he longiudinal profile from he convex in he evolvenoid one o he concave in he hyperboloid one lead o ha he ranveral (radial) ocillaion are going on in he oppoie phae in hee one. In a reul he above plae i a plae of an acion of he radial cyclic hear re, promoing a haking of he pecimen maerial rucure. In hi cae he above ilence bel render he negaive influence on he pecimen rengh. Revering o he bae objec of he given aricle, i i alo neceary o noe, ha he necking i he ypical phenomenon for uch meal a he mild eel. A he ame ime he feaure of he meal inernal rucure lead no only o he differen modificaion of he necking, bu o a fracure wihou he necking. For example, he cylindrical pecimen of he grey iron under acion of an axial enion force i fracured wihou he viible reidual deformaion (he quare-break fracure). Such 8

9 diincion of he pecimen behavior i explained by ha he ulimae rengh of he grey iron i in 5 ime le han i compreion rengh. Therefore, in conra o he eel pecimen he grey iron pecimen fracure i going on by mean of a forming of he ranveral hoop crack on i exernal urface and i ubequen fa developing o he pecimen axi. Suppoe o, wih he aking ino accoun of he above menioned Bridgman experimen, ha he grey iron pecimen will be fracured wih he necking under acion of a combinaion of he ufficienly high hydroaic preure and he comparaively no grea axial compreive force. Nex queion, affeced in he end of he Soluion ecion of he given aricle on pure uniaxial reed ae ha he grea mehodological ignificance and one require he more deailed explanaion. The fac i ha he radiional coure of Srengh of Maerial from he very oue preen he rod enion a a imple enion. The deerminaion of a reed ae in i i rericed by a diviion of he enion force by he rod cro-ecion area. Thi imple enion i hen preened a he uniaxial reed ae. Furher he radiional approach enviage a udy of he planar reed ae and hen he riaxial (3-D) reed ae a a complex reed ae. In ha way he radiional eaching i formally conruced righly - from a imple o a complex. A he ame ime he reul of he enion e of he hin wire of he mild meal (wih he wire diameer, compared wih he grain diameer of he wire meal) how ha in hee cae he fracure i going on wihou he necking (ee, for example, I. J. Dechjar` aricle in he above menioned he Colleced aricle [3]). Thi feaure in a comparion wih he reul of he enion e of he andard pecimen of he mild eel eifie o ha he rod wih a diameer 10 0 mm i preened in he kind of a bunch of he hin wire wih he oal cro-ecion area, equal he olid rod cro-ecion area. In a reul hoe, who had good udied he Srengh of Maerial, one are aonihed by he necking and hoe, who had weakly udied hi coure, one do no hink abou i. Thi paradoxal iuaion exi more han 100 year. In hi connecion i i appropriae o appeal o prof. L. R. G. Treloar` book [18], in which he auhor adduce an example of a enion of he uual polyehylene film rip 10 cm long and 1 cm widh. In hi experimen he enion i accompanied by houlder effec in he kind of he local narrowing of he rip, which envelop gradually all i enioned par imilar o he local narrowing, enveloped he all lengh of he mild eel rod, enioned on he iniial age of i plaic deformaion. Bu, in conra o he eel rod, he polyehylene rip fracure i going on wihou he necking, i. e. pracically brile. Such fracure of he very plaic maerial i ipulaed by he following. On he iniial age of he plaic deformaion he polyehylene molecular chain raighen oneelf and one are reched along he enion direcion, loing a par of i ranveral ie. In a reul of i he enire maerial i ranformed pracically ino a bunch of he microfibre, he every of which and all one ogeher are fracured wihou he necking. Such enire maerial i more imilar o he wire rope han he olid rod. Thi i wha, in brief, follow from he problem oluion on he imple enion and on he necking. In he dicuion concluion i i neceary o noe ha he ucceive oluion of he aed problem i he nex in urn example of a uing of he Phyical Enemble mehod, developed by auhor of he given aricle. An effecivene of he mehod for a phyically adequae and mahemaically ufficienly ric conrucion of he model of he ome phenomena in a field of he fluid dynamic wa howed in previou aricle [19], wrien by he auhor eparaely, and in aricle [0 8], wrien by he auhor wih hi collaboraor. 9 Final remark The approach, he problem oluion and he conidered example, aed in given aricle, creae he neceary prerequiie for he fruiful oluion of he problem and queion, bound wih an analyi of he rained and reed ae of he olid and liquid on he new unied phyically enible mehodological bae.

10 10 Acknowledgemen Auhor wan o expre hi profound repec o he ouanding precuror, uppoing ha he modern reul, filling he mechanic eparae gap, are ipulaed by he naural developmen of he grea heriage. Auhor wan o expre hi deep graiude o hi on Alexey for yping of hi aricle ex and carrying ou of hi aricle graphical par. Auhor wan alo o expre hi deep graiude o hi daugher Caherina for her caring for her faher. Auhor dedicae hi aricle o he unforgeable memory of Nina Sepanovna Nikienko he alened peciali, lecurer on Srengh of Maerial a Polyechnic Iniue of Odea, and [1] W. Kune und G. Sach. ZS. V.D.I., 7, 1011, (198) [] D. Pedoe, Geomery and he Liberal Ar, Penguin Book Ld., Harmondworh, (1976) Tranl. from Engl. ino Ru., MIR Publihing, Mocow, (1979) [3] V. G. Oipov, Proce of Fracure by Shear a Simple Compreion and Tenion, in Colleced aricle on Srengh of Meal, USSR Academy of Science Publihing, Mocow, (1956) [4] W.Köer, Arxiv für angewade Eienhüenween, Gruppe E, 47, (199) [5] E. Orowan, ZS. Phyik, 89, 634, (1934) [6] M. Polanyj, ZS. Phyik, 89, 660, (1934) [7] G. I. Taylor, Proc. Roy. Soc. (London), A 145, 36, (1934) [8] Lueder, Dingler Polyechn. Journ., (1854) [9] E. Bollenrah, V. Hauk und E. Owald, ZS., V.D.I., 83, 5, 19, (199) [10] S. P. Timohenko, Srengh of Maerial, Par II, D. Van Norand Co., Inc., Princeon, NJ, (1956) - Tranl. from Engl. ino Ru., Nauka Publihing, Mocow, (1965) [11] E. A. Davi (Weinghoue Reearch Laboraorie), J. Appl. Mech., Vol. 1, p. 13, (1945) and Vol. 15, p. 16, (1948) [1] P. W. Bridgman, Sudie in Large Plaic Flow and Fracure, McGraw-Hill, New York, (195) [13] V. I. Feodojev, Seleced Problem and Queion on he Srengh of Maerial Nauka Publihing, Mocow, (1967) [14] Y. G. Panovko, I.I. Gubanova, Sabiliy and Ocillaion of Elaic Syem, Nauka Publihing, Mocow, (1987) [15] F. H. Newman, V. H. L. Searle, The General Properie of Maer, Edward Arnold & Co., London, (1948) [16] J.W. Sru - Baron Rayleigh, Theory of Sound, Vol. 1, nd Ed., McMillan and Co. Ld., London, (196) Tranl. ino Ru., Sae Publihing of Engineering and Theoreic Lieraure, Mocow Leningrad, (1940) [17] The Grea Sovie Encyclopedia, Vol. 1, Sovie Encyclopedia Publihing, Mocow, (1973) [18] L. R. G. Treloar, Inroducion o Polymer Science, Wykeham Publicaion (London) Ld., London & Wincheer, (1970) - Tranl. from Engl. ino Ru., MIR Publihing, Mocow, (1973) [19] S.L. Arenjev To he fluid moion dynamic, hp://uk.arxiv.org/ab/phyic/ , 006. [0] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, Laval Turbonole Dynamic, hp://uk.arxiv.org/ab/phyic/ , 005. [1] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, The boundary and coninual ranfer phenomena in fluid and flow, hp://uk.arxiv.org/ab/phyic/ , 003. [] S.L. Arenjev, I.B. Looviki, Y.P. Sirik, The Gadynamic Fir Problem Soluion. The Ga Sream Parameer, Srucure, Meering Characeriic for Pipe, Nole, hp://uk.arxiv.org/ab/phyic/ , 003. [3] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, The laminar flow inabiliy crierion and urbulence in pipe, hp://uk.arxiv.org/ab/phyic/ , 003 [4] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, The ga equaion for ream, hp://uk.arxiv.org/ab/phyic/ , 003

11 [5] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, The flowing yem gadynamic.par 4: Influence of he inciden flow velociy on he ouflow velociy ou of flowing elemen, hp://uk.arxiv.org/ab/phyic/030083, 003 [6] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, The Flowing Syem Gadynamic Par 3: Sain- Venan Wanel formula modern form, hp://uk.arxiv.org/ab/phyic/03008, 003. [7] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, The Flowing Syem Gadynamic Par : Euler momenum conervaion equaion oluion, hp://uk.arxiv.org/ab/phyic/03000, 003. [8] S.L. Arenjev, I.B. Looviki, Y.P.Sirik, The Flowing Syem Gadynamic Par 1: On aic head in he pipe flowing elemen, hp://uk.arxiv.org/ab/phyic/ ,

12 1 a) P b) c) N = P / N = P / r 1 r A A = A/ A c = A/ r 1 = P / A = c c = N / A = N / A = = 1. 5 e) = d) r = = p c r = f) g) = p c = = p r = p' c r = p' c = p' = p' = 3p' Fig.1. Diagram of he force and ree for he cylindrical rod under acion of he axial enion force: a,b,c) diagram of he force and he field line, acing in he rod and i par hell (upercrip ) and core (upercrip c) correpondingly; d,e) diagram of he ranveral hoop and radial ree in he rod hell and in he rod core, correpondingly, and i muual preure in he iniial age of he rod plaic deformaion; f,g) diagram of he ranveral hoop and radial ree in he rod hell and in he rod core, correpondingly, and i muual preure in he neck one

13 13 R α Fig.. Diagram of a conrucing of he one-hollowed hyperboloid and of he longiudinal profile of he enioned cylindrical rod on he necking age under acion of he axial enion force only

14 14 A B C E K L ` Fig. 3. Diagram of a conrucing of he longiudinal profile of he enioned cylindrical rod on he necking age under acion only of he laeral hydroaic preure: he developmen angle of he evolven reache he maximum quaniy `; AB he hyperboloidal core; BC he evolveniodal bel; CE he orhogonal hyperboloid bel; KL he orhogonal hyperboloid in he hyperboloidal core

15 15 Ri R e Ae A i = 3 R e A e = 1 = ` Silence bel a) b) Fig. 4. The conrucion diagram of he middle urface profile of he bell (he Tar-bell in he Mocow Kremlin): a) combinaion of hyperboloid and cone; b) combinaion of hyperboloid, cone and oru