myari deformadi sxeulis meqanikis ganviTarebis etapebi XX ...

aleqsandre daTuaSvilimyari deformadi sxeulis meqanikisganviTarebis etapebi XX saukuneSiწარმოდგენილია დოქტორის აკადემიური ხარისხისმოსაპოვებლადსაქართველოს ტექნიკური უნივერსიტეტითბილისი, 0175, საქართველოთვე, 2008 წელი© საავტორო უფლება 2008, aleqsandre daTuaSvilii

საქართველოს ტექნიკური უნივერსიტეტიsamSeneblo fakultetiჩვენ, ქვემორე ხელისმომწერნი ვადასტურებთ, რომ გავეცანითaleqsandre daTuaSvilis მიერ შესრულებულ სადისერტაციო ნაშრომსდასახელებით: `myari deformadi sxeulis meqanikis ganviTarebisetapebi XX saukuneSi~ და ვაძლევთ რეკომენდაციას საქარველოსტექნიკური უნივერსიტეტის samSeneblo fakultetis სადისერტაციოსაბჭოში მის განხილვას დოქტორის აკადემიური ხარისხის მოსაპოვებლად.თარიღიხელმძღვანელი:Tamaz bacikaZeრეცენზენტი:რეცენზენტი:რეცენზენტი:ii

საქართველოს ტექნიკური უნივერსიტეტი2008 წელიავტორი: aleqsandre daTuaSviliდასახელება: myari deformadi sxeulis meqanikis ganviTarebisetapebi XX saukuneSiფაკულტეტი : samSeneblo fakultetiხარისხი:დოქტორისხდომა ჩატარდა: თარიღიინდივიდუალური პიროვნებების ან ინსტიტუტების მიერზემომოყვანილი დასახელების დისერტაციის გაცნობის მიზნით მოთხოვნისშემთხვევაში მისი არაკომერციული მიზნებით კოპირებისა და გავრცელებისუფლება მინიჭებული აქვს საქართველოს ტექნიკურ უნივერსიტეტს.ავტორის ხელმოწერაავტორი ინარჩუნებს დანარჩენ საგამომცემლო უფლებებს და არცმთლიანი ნაშრომის და არც მისი ცალკეული კომპონენტების გადაბეჭდვა ანსხვა რაიმე მეთოდით რეპროდუქცია დაუშვებელია ავტორის წერილობითინებართვის გარეშე.ავტორი ირწმუნება, რომ ნაშრომში გამოყენებული საავტოროუფლებებით დაცულ მასალებზე მიღებულია შესაბამისი ნებართვა (გარდაიმ მცირე ზომის ციტატებისა, რომლებიც მოითხოვენ მხოლოდ სპეციფიურმიმართებას ლიტერატურის ციტირებაში, როგორც ეს მიღებულიასამეცნიერო ნაშრომების შესრულებისას) და ყველა მათგანზე იღებსპასუხისმგებლობას.iii

რეზიუმეmyari deformadi sxeulis meqanika dReisaTvis erT-erTiwamyvani dargia, romelic gamoiyeneba TiTqmis yvela sainJinromimarTulebaSi. gansakuTrebiT kargadaa damuSavebuli drekadobisTeoriis is nawili, sadac mecnierebis miRwevebi dainerga XXsaukuneSi Txelkedliani sivrciTi konstruqciebis gamoyenebiTmanqanaTmSeneblobaSi, aviamSeneblobaSi, gemTmSeneblobaSi,nagebobaTa mSeneblobaSi da teqnikis sxva dargebSi, asevesamrewvelo, sasoflo-sameurneo, savaWro, kulturul-sportulinagebobebis gadaxurvis racionaluri gadawyvetis saqmeSi.naSromSi damuSavebulia drekadobis Teoriis sivrciTiamocanebis istoria. brtyeli amocanebis amoxsnis meTodebis damisi ganviTarebis istoria, teqnikis sxvadasxva dargSi myarideformadi sxeulis meqanikis gamoyenebis istoria daganviTarebis etapebi meoce saukuneSi.gadmocemulia drekadobis wrfivi Teoriis istoria.aRniSnulia, rom drekadi tanis wonasworobis amocanaSi,rodesac ar arseboben masaTa Zalebi, moinaxeba zogadigamosaxulebebi, romlebic SesaZleblobisdagvarad akmayofilebenCveulebriv diferencialur gantolebebs da ise arian agebuli,rom drekadobis Teoriis gantolebebi sruldebodes amgantolebebis ZaliT. drekadobis Teoriis gantolebis amoxsnisaseTi warmodgena mocemuli iqna p. papkoviCis mier da ramdenadmegvian g. neiberis mier.miCneulia, rom amonaxsnSi, iseve rogorc zogadi amonaxsnebissxva formebSi, saWiroa vxedavdeT drekadobis TeoriissasazRvro amocanebis amoxsnis sasargeblo damxmare saSualebas,romelic dauSvebs laplasis gantolebebis klasikuri kerZoamonaxsnebis uSualo gamoyenebas. konkretuli amocanis amoxsnisagebisas meoTxe harmoniuli funqciis SenarCuneba aadvilebs amamonaxsenTa arCevas, amitom ar saWiroebs masze uaris Tqmas.aRwerilia sen-venanisa da almanzis amocanebis istoria.rogorc cnobilia, prizmuli Reros Tavisufali grexis Sesaxebamocana daiyvaneba harmoniul problemaze, romlis amoxsnaTameTodebi kargadaa SemuSavebuli. adreuli naSromebi Reroebisgrexaze miZRvnilia am amocanis Sekruli saxiT antrigonometriuli mwkrivebis saSualebiT amoxsnisadmi; maTmiekuTvneba b. galiorkinis statiebi, romlebSic gamokvleuliaprizmis grexa, romelsac tolferda marTkuTxa samkuTxedissaxis ganivi kveTi aqvs (1919) da paraboluri ganivi kveTisprizmebi (1924); amocana rigi kveTebis grexis Sesaxeb, romlebicSemosazRvrulia algebruli mrudebiT, gadawyvetilia d. benovis(1935, 1937) da d. gavras (1939) naSromebSi; mogvianebiT paraboluriprizmebis grexiT dakavebuli iyo v. bloxi (1959). mTliani an Rrulilvebis grexisas radialuri bzaris gavlena Seswavliliaa.lokSinis (1928) da v. liskovis (1930) statiebSi. grexis Teoriisiv

amocanebis amoxsnis sxvadasxva meTodebs (eqsperimentalurimeTodebis CaTvliT) moicavs a. dinikis monografia (1938).1925 wels g. kolosovma da d. gavrma grexis amocanisamoxsnisas pirvelad gamoiyenes kompleqsuri cvladebi; maTganixiles amocana mcire centraluri kuTxiT arawriuliseqtoris grexis Sesaxeb. am mimarTulebiT fundamentaluriSedegebi miiRo n. musxeliSvilma (1929), romelmac aCvena, romcaladbmuli da oradbmuli midamos grexis amocana, romelicasaxavs mocemul midamos Sesabamisad wresa da wriul rgolze,daiyvaneba kompleqsuri cvladis funqciis moZebnaze. prizmuliReroebis grexis amocanebis amoxsnisas kompleqsuri cvladisfunqciis Teoriis meTodebs iyenebdnen kvleviT muSaobaSi d.avazaSvili (1940), a. batirevi (1953), x. muStari (1938), a.ugodCikovi(1956) da sxva mecnierebi.r. kuzminma (1946) gamoiyena konformuli anasaxi sxvaformaSi; man dasagrexi Reros sixistis uSualo gamoTvlisaTvismiiRo moxerxebuli formula, romelmac saSualeba mogvcagamogveTvala sixiste im profilebisaTvis, romelTa konturiSeicavs kuTxis wertilebs. SemdgomSi am sakiTxebze muSaobdap.kufariovi, romlis meTodi o. babakovma (1954) gamoiyena zeturiprofilis grexis ganxilvisas.maTematikuri drekadobis Teoriis Sereul amocanebadCveulebriv drekadi wonasworobis iseT amocanebs gulisxmoben,roca sxeulis zedapirze ganlagebulia sxvadasxvagvari tipissasazRvro pirobebis gamyofi xazebi. Tu gansaxilveli drekadisxeulis zedapiri Sedgeba ramdenime gluvi waxnagisagan, maSinSeiZleba warmogvidges Sereuli amocanis xarisxobrivadgansxvavebuli ori ZiriTadi varianti.momdevno wlebSi drekadobis Teoriis zogadi gantolebebisgamoyenebaze da kerZod, papkoviC-neiberis funqciis gamoyenebazedafuZnebuli meTodebis ganviTarebam, SesaZlebloba mogvcanaxevarsivrcis drekadi wonasworobis bevri saerTo Sereuliamocana dagveyvana potencialis Teoriis Sereuli amocanebiszogierT klasebamde. potencialis Teoriis msgavsi amocanebismeTodurma damuSavebam saSualeba mogvca zustad amogvexsnazogierTi sakontaqto da msgavsi Sereuli amocanebi. ammeTodebidan ZiriTadebi arian: sferoidaluri da elifsoidalurikoordinatebis gamoyeneba (a. lurie); grinis funqciis ageba dagamoyeneba (l. galini, m. leonovi, 1953); integralur gantolebaTameTodi (i. Staermani, v. mosakovski, 1953); toroidulikoordinatebisa da integraluri gardaqmnebis gamoyeneba(i.ufliandi 1956, 1957); kompleqsuri potencialebis meTodi(m.rostovcevi, 1953, 1957).damuSavebulia Txelkedlian sivrciT sistemebSi myarideformadi sxeulis meqanikis ganviTarebis istoria.konkretuli Sinaarsis pirveli Sedegebi, romlebicganekuTvneba brtyeli profilebis wonasworobas miRebuli iyog.kolosovisa da n.musxeliSvilis mier.v

n. musxeliSvilma mogvca pirveli da meore ZiriTadiamocanebis martivi amonaxsni wris, wriuli rgolisa dausasrulo sibrtyisTvis wriuli naxvretiT. damuSavebuli iqnamravali kerZo magaliTi sxvadasxva saxis gare zemoqmedebebisaTvis.amgvari saxis areebisaTvis ar moiTxoveba winaswari konformuliasaxva. gamoiyena ra konformuli anasaxi musxeliSvilmagadawyvita im droisaTvis rTuli amocana mTliani elifsiswonasworobis Sesaxeb.xarisxovani mwkrivebis meTodiT iyo gamokvleuli amocanaTanafokusuri elifsuri rgolis Sesaxeb (a. kalandia, 1953). amamocanis efeqturi amoxsnis algoriTmi kidev ufro adremiTiTebuli iqna m. Seremetievis mier, romelic iyenebdafunqcionaluri gantolebebis meTods konformul asaxvebTannaerTSi.Catarebul gamokvlevaTa istoriuli mimoxilvisa daanalizis safuZvelze SeiZleba gavakeToT Semdegi daskvnebi:− samecniero literaturaSi praqtikulad ar aris ganxiluliTxelkedliani sivrciTi sistemebis gaangariSebis meTodebisistoriuli analizi;− zogierTi Sromebis gamoklebiT, ganzogadoebuli impulsurifunqciebi gamoiyenebian mxolod diferencialuri gantolebebisCawerisaTvis, magram ara maTi amoxsnebis misaRebad. amave droswyvetili funqciis SemoReba iZleva gaangariSebis iseTi,principulad axali meTodebis miRebis saSualebas, romlebicafarToeben amoxsnadi amocanebis klass da anzogadeben yvelaamoxsnad amocanebs erTian meTodologiur safuZvelze.drekadobis wrfivi Teoriis istoriis safuZvelze dReisaTviskidev ufro meti istoriuli masalis gadmocema da misianalizia Sesasrulebeli;− sen-venanisa da almanzis amocanebSi prizmuli RerosTavisufali grexis amocana daiyvaneba harmoniul problemaze,romlis amonaxsnTa meTodebi kargadaa damuSavebuli dagaanalizebuli;− drekadi prizmuli Zelebis daZabul-deformirebulimdgomareobis dadgena, rodesac Zelis boloebze moqmedebsnebismier ZalTa sistema, aseve warmoadgens drekadobisTeoriis erT-erT ZiriTad da rTul maTematikur amocanas.maTematikuri TvalsazrisiT igi ar aris bolomdegadawyvetili, Tumca e.w. `sen-venanis principis~ daxmarebiTxerxdeba am amocanis gadawyveta, romelic miaxloebiTia da arSeiZleba CaiTvalos zustad. swored amgvar klasikurmidgomadaa CaTvlili sen-venanis mosazreba, romelicliteraturaSi damkvidrda `sen-venanis principis~, kerZod kisen-venanis naxevrad Sebrunebuli meTodis saSualebiT.− musxeliSvilis gamokvlevebi problemebis farTo klass moicavs,misma Sromebma didi gavlena moaxdines meqanikisa damaTematikis mTeli rigi mimarTulebebis Semdgom ganviTarebaze.vi

SummaryNowadays the mechanics of elastic solid body is one of the leading fields. It iswidely used in all engineering sciences.The section of the theory of elasticity is particularly well developed wherescientific achievements have been implemented in the XX-th century, in particularthin walled space constructions were used in aircraft building, mechanicalengineering, ship building, civil engineering and in other fields of technique and alsoin roof constructions of industrial, agricultural, trade, entertaining and sport building.The present work concerns the history of spatial problems of the theory ofelasticity, as well as, the history of plane problems solution and its development, thehistory of using the mechanics of elastic solid body in various scientific fields and thestages of its development in the XX-th century.The history of the theory of linear elasticity is given. It is noted that in theproblem of solid body equilibrium, in the absence of mass forces, general expressionscan be found which, as far as possible, satisfy common differential equations and areset up in such a way that the equations of elastic theory be solved by these equations.Such representation of the solution of the equation of the theory of elasticity wasgiven by P.Papkovich and later by G.Neibere.We think that in solution, as well as, in other forms of general solutions wehave to presume the useful additional means of solution of boundary problems of thetheory of elasticity which will allow using of classical partial solutions of Laplaceequation. The preservation of the fourth harmonious function when constructing thesolution of the concrete problem simplifies their choosing, therefore it should not beneglected.By using the theory of elasticity in isotropic solid we come to Galiorkin-Busineski solution.The work also describes the history of Sen-Venan and Almanz problems. As itis well known the problem of prismatic rod twist is reduced to harmonious problemthe solution methods of which are well developed. Early works about rod twistconcerns the solution of this problem in closed form or with trigonometrical series.Among them are the articles by B.Galiorkin, in which twisting of prisms having crosssectionof isosceles right-angled triangle form (1919) and prisms with parabolic crosssections(1924) are investigated. The problem about twisting of some cross-sectionslimited by algebraic curves was solved in the works by D.Benov (1935, 1937) andD.Gavre (1939); later the problem of parabolic prisms was investigated by V.Block(1959). The study of the effect of radial crack at twisting of solid and hollow shaftswas given in the works by A.Lokshin (1928) and V.Liskov (1930). Also, A.Dinnik’smonograph is dedicated to different methods (including experimental ones) of solutionof twisting theory problems.In 1925 G.Kolosov and D.Gavre were the first who used complex variableswhen solving the problem of twisting. They considered the problem of noncircularsector twisting by small central angle. In this field fundamental results were obtainedby N.Muskhelishvili (1929) who discovered that the problem of twisting of singleanddouble-brace medium which represents the given medium on circle and circularring, respectively, is reduced to searching of the function of complex variable. Whensolving the problems of prismatic bar twisting the methods of complex variablefunction theory were used for different profiles of such bars by D.Avazashvili (1940),A.Batirev (1953), Kh.Mushtar (1938), A.Ugodchikov (1959) and others.vii

R.Kuzmin (1946) used the conformal image in other form. He was the firstwho received a convenient formula for direct calculation of twisted bar rigidity whichallowed to calculate rigidity for the profiles, the contour of which contain anglepoints. Later, P.Kufariov was working on this problem whose method was used byO.Babakov (1954) while considering twisting of z-profile.When discussing mathematical elasticity problems, we generally mean suchproblems of elastic equilibrium when different type lines separating boundaryconditions are distributed on the surface of the body. If the surface of the above givenelastic body consists of several smooth faces then qualitatively differing two mainversions of the mixed problem are received.In the following years the development of the methods based on using ofcommon equations of the theory of elasticity, particularly on using of Papkovich-Neibere function enabled to reduce many general mixed problems of half-space elasticequilibrium to some classes of mixed problems of potential theory. The elaboration ofthe problems analogous to potential theory allowed precise solution of some contactand complex problems. The basic of these methods are: use of spheroidal andellipsoidal coordinates (A.Lourie); plotting up and use of Green’s function (L.Galin,M.Leonov, 1953); method of integral equation (I.Shtaermann, V.Mosakovski, 1953);use of toroidal coordinates and integral transformations (I.Yuflyand, 1956, 1957);method of complex potentials (M.Rostovtsev, 1953, 1957).The history of development of the mechanics of solid elastic body in thinwalled spatial systems is given.The first concrete results in equilibrium of plane profiles were received byG.Kolosov and N.Muskhelishvili.N.Muskhelishvili elaborated the simple solution of the first and the secondmain problems for circle, circular ring and infinite plane with circular hole. A numberof particular examples have been worked out for different external effects. For suchareas preliminary conformal depiction is not necessary. Using conformal depictionN.Muskhelishvili solved a very complicated, for that time, problem about equilibriumof continuous ellipse.By the method of qualitative series the problem of co-focus elliptic ring wasinvestigated (A.Kalandia, 1953). Earlier the algorithm of effective solution of thisproblem was given by M.Sheremetiev who used the method of functional equationstogether with conformal depiction.On the basis of historical review and analysis of the carried out researches thefollowing solutions can be done:− In scientific literature there practically is not considered the historical analysis ofcalculation methods of thin walled spatial systems;− Except some works, the generalized pulse functions are used only for writingdown differential equations but not for their solution. At the same time theintroduction of interrupted function gives essentially new methods of calculationwhich widen the class of solvable problems and generalize them on uniformmethodological basis. On the basis of the history of linear elasticity theory muchmore historical material can be compiled and analyzed.− In the problems of Sen-Venan and Almanz the problem of free twist of prismaticbar is reduced to harmonious problem, the methods of their solution being welldeveloped and analyzed.− The establishment of stress-strained state of elastic prismatic beams when arbitraryforce system is acting upon the beam ends, also represents one of the basic andcomplicated mathematical problem of the theory of elasticity. From theviii

mathematical point of view it is not solved to the end, though by means of the socalled“Sen-Venan Principle” we can solve this problem which is approximatedand cannot be considered as precise. The idea by Sen-Venan is considered asclassical approach which in literature is establishes by means of “Sen-Venanprinciple”, particularly, by semi inverted method of Sen-Venan.− The researches by Muskhelishvili contain a wide class of problems. His works hada great influence on further development of a number of directions of mathematicsand mechanics.ix

შინაარსიSesavali .......................................................................................................................................... 131. literaturis mimoxilva1.1. drekadobis wrfivi Teoriis istoria ............................................... 161.1.1. zogadi amonaxsnebi da arsebobis Teoremebi .......................... 161.1.2. ZabvaTa funqciis tenzori ......................................................................... 211.1.2.1 sivrciTi amocanis integraluri gantolebebi .................. 261.1.2.2. robenis elastostatikuri amocana .............................................. 311.1.3. drekadobis Teoriis sivrciTi amocanebis Sesaxeb ...... 331.1-is daskvnebi ................................................................................................................... 472. Sedegebi da maTi gansja2.1. sen-venanisa da almanzis amocanebis warmodgena ..................... 482.1.1. grexisa da Runvis amocanebis dasma da gadawyveta ...... 552.1-is daskvnebi .................................................................................................................. 632.2. drekadi tanis statikis Sereuli sivrciTiamocanebis Seswavlis istoria .............................................................. 642.2.1. drekadobis Teoriis brtyeli amocanebis dasmada amoxsnis meTodebi .................................................................................. 762.2.1.1. brtyeli amocanis amoxsnis zogadi kompleqsuriwarmodgena ...................................................................................................... 762.2.2. drekadobis Teoriis brtyeli ZiriTadi amocanebisformulireba ................................................................................................. 792.2.3. brtyeli amocanebis amoxsnis meTodebi .................................... 842.2.3.1. funqciis holomorfuloba (analizuroba) ......................... 842.2.3.2. mocemuli funqciis konformuli asaxva ............................... 882.2.3.3. koSis tipis integralebis gamoyeneba ...................................... 902.2.3.4. mravalbmul areebSi kompleqsuri warmodgena ............... 912.2.3.5. brtyeli amocanis amoxsnis xerxebi .......................................... 952.2.3.6. lauriCela-Sermanis gantolebis gamoyeneba ..................... 972.2.3.7. helderis pirobis gamoyeneba ............................................................ 982.2-is daskvnebi ................................................................................................................. 1042.3. myari deformadi sxeulis meqanikis ganviTarebisistoria Txelkedlian sivrciT sistemebisaTvis .............. 105x

2.3.1. brtyeli drekadobis Teoriis amocanebisgamokvlevebis ZiriTadi Sedegebi ................................................. 1052.3.2. uban-uban erTgvarovani garemo. Semagrebuli dagaZlierebuli firfitebi ..................................................................... 1152.3.3. Sereuli da sakontaqto amocanebi .............................................. 1232.3.4. anizotropuli tanis drekadobis Teoriis brtyelistatikuri amocana ........................................................................................ 1252.3.5. sivrciTi maRali Zabvis eleqtrogadamcemixazebis konstruqciebis qarsawinaaRmdegomdgradoba .............................................................................................................. 1292.3.6. bzarebis mqone myife sxeulebis wonasworoba ................ 1372.3.7. myife da plastikuri masalebis rRvevisproblemebi ............................................................................................................ 1392.3-is daskvnebi ............................................................................................................... 1453. daskvna ..................................................................................................................................... 146gamoyenebuli literatura ....................................................................................... 148xi

ნახაზების ნუსხაnax. 1. nagebobaTa sakuTari rxevebis intervalTamiaxloebiTi sidideebi ............................................................................ 135nax. 2. qaris siCqaris saangariSo da realuri epiurebi ........ 136nax. 3. temperaturisa da deformaciis siCqaris gavlenisgrafikuli asaxva tyviis masalisagan damzadebulinimuSis plastikurobaze gaWimvisas ............................................ 144xii

Sesavalimyari deformadi sxeulis meqanika dReisaTvis erT-erTi wamyvanidargia, romelic gamoiyeneba TiTqmis yvela sainJinro mimar-TulebaSi da didad mniSvnelovania misi ganviTarebisaTvis.gansakuTrebiT kargadaa damuSavebuli drekadobis Teoriis isnawili, sadac mecnierebis miRwevebi dainerga XX saukuneSi,kerZod Txelkedliani sivrciTi konstruqciebis gamoyenebiTmanqanaTmSeneblobaSi, aviamSeneblobaSi, gemTmSeneblobaSi,nagebobaTa mSeneblobaSi da teqnikis sxva dargebSi, asevesasoflo-sameurneo, savaWro, kulturul-sportuli nagebobebisgadaxurvis racionalurad gadawyvetis saqmeSi. Tanamedrovedidmaliani nagebobebis mSenebloba da maRali simtkicismaxasiaTeblebis mqone mciremoduliani masalebis praqtikaSidanergvas mivyavarT Txelkedliani konstruqciebis gaangariSebisdros sisqesTan SedarebiT sakmaod didi CaRunvebis gaTvaliswinebisaucileblobamde. mrewvelobisa da mSeneblobis sxvadasxvasferos ganviTareba dakavSirebulia arsebulis daxvewasa da iseTiaxali konstruqciebis SeqmnasTan, romlebic Seicaven: gisosebs,firfitebs da Reroebs, SemagrebiT, texvebiT, xvretebiT,WrilebiT, wertilovani sayrdenebiT.konstruqciebSi geometriuli da fizikuri parametrebisararegularoba iwvevs Zabvebis mniSvnelovan koncentracias daqmnis bzarebis an plastikuri deformaciebis gavrcelebis saSiSzonebs. umetes SemTxvevebSi maTi mzidunarianoba ganisazRvrebasimtkicis pirobebiT an Zabvebis mdgradobis dakargviTkoncentraciis zonaSi. regularobis darRvevis sxva saxeebsganekuTvneba zedapiris texva, rasac adgili aqvs naoWovan damravaltalRovan gadaxurvebSi. daZabul mdgomareobaze gavleniTisini wiboebis analogiurni arian.regularobis darRvevis adgilebSi ZabvaTa koncentraciiszonebi (wibos bolo, diskretuli bmebi) arsebiT gavlenasaxdenen Txelkedliani konstruqciebis zidvis unarze da13

mdgradobaze. am dros tradiciuli analizebi da ricxviTimeTodebi wibovani Txelkedliani konstruqciebis daZabuldeformirebulimdgomareobis gamosakvlevad naklebad efeqturia.amitom saWiro xdeba maRali klasis amocanebis amoxsnis axaliefeqturi meTodebis SemuSaveba. swored zemoTaRniSnulis gamoaucilebelia myari deformadi sxeulis meqanikis zedmiwevniTiSeswavla da misi ganviTarebis istoriis codna, romelic metadaqtualuria. winamdebare naSromi exeba am sakiTxebis ganviTarebisetapebs meoce saukuneSi, roca es mecniereba gansakuTrebulaRmavlobas ganicdida.aqvea gadmocemuli drekadobis Teoriis sivrciTi amocanebisganxilva, sen-venanisa da almanzis amocanebi, drekadi tanisstatikis Sereuli sivrciTi amocanebi, brtyeli amocanebisamoxsnis meTodebi da misi ganviTarebis etapebi, kompleqsuricvladis funqciis Teoriis meTodebi da misi ganviTarebisetapebi meoce saukuneSi, brtyeli drekadobis Teoriis amocanebisgamokvlevebis ZiriTadi Sedegebi.damuSavebulia drekadobis Teoriis sivrciTi amocanebiskvlevis istoria. brtyeli amocanebis amoxsnis meTodebis da misiganviTarebis istoria, teqnikis sxvadasxva dargSi myarideformadi sxeulis meqanikis gamoyenebis istoria daganviTarebis etapebi meoce saukuneSi.naSromis praqtikuli Rirebuleba mdgomareobs imaSi, rommyari deformadi sxeulis meqanikis ganviTarebis etapebs meocesaukuneSi, romelic gadmocemulia istoriul doneze, didisargebloba eqneba am sakiTxebiT dainteresebulTa Soris, radganmisi gamoyeneba SesaZlebelia fizika-maTematikisa da teqnikisganviTarebis nebismieri dargis specialistebisTvis, romlisrealizebac aucilebelia aRniSnul dargebSi.Sedegebis efeqturoba ganpirobebulia, imiT, rom yvelaaRniSnuli sakiTxi eyrdnoba meoce saukuneSi gamoqveynebul sxvadasxvaavtoris mier miRebuli istoriuli masalidan, aseve14

mravali saarqivo masalidan da damakmayofilebelia TanadamTxveviT.naSromis aprobacia. disertaciis ZiriTadi Sedegebi moxsenebuliqna:− saqarTvelos teqnikuri universitetis masalaTa gamZleobisa dadrekadobis Teoriis kaTedris samecniero seminarze (Tbilisi,2005 - 2006w.);− saqarTvelos teqnikuri universitetis samSeneblo meqanikis daseismomedegobis samecniero seminarze (Tbilisi, 2006w.);− saqarTvelos teqnikuri universitetis masalaTa gamZleobisada drekadobis Teoriis, Teoriuli meqanikis, samSeneblomeqanikis da seismomedegobis kaTedrebis gafarToebulsamecniero seminarze (Tbilisi, 2006w.);− saqarTvelos teqnikuri universitetis samSeneblo meqanikisada nagebobaTa seismomedegobis, masalaTa gamZleobisa dadrekadobis Teoriis da xidebisa da gvirabebis kaTedrebisgafarToebul sxdomaze (Tbilisi, 2006w. oqtomberi).15

1. ლიტერატურის მიმოხილვა1.1. drekadobis wrfivi Teoriis istoria1.1.1. zogadi amonaxsnebi da arsebobis Teoremebidrekadi tanis wonasworobis amocanaSi, rodesac ararseboben masaTa Zalebi, moinaxeba zogadi gamosaxulebebi(gadaadgilebebis an Zabvebis), romlebic SesaZleblobisdagvaradakmayofileben martiv diferencialur gantolebebs da ise arianagebuli, rom drekadobis Teoriis gantolebebi sruldebodes ammartivi gantolebebis ZaliT. `martivTa~ rols TamaSobenlaplasisa da biharmoniuli gantolebebi; amasTan sasurveliafunqciaTa umciresi ricxvi. zogadi amonaxsnebis codnasaSualebas iZleva drekadobis Teoriis gantolebebis kerZoamonaxsnebis Sedgenisas gamoviyenoT kargad cnobili `katalogebi~`martivi~ gantolebebis amonaxsnebisa ama Tu im koordinatebSi;drekadobis Teoriis sasazRvro amocanebi, umartivesTa(naxevarsivrce, brunviTi tanis grexa da sxva) gamoklebiT, verdaiyvaneba, dirixlesa da neimanis tipis amocanebze laplasisgantolebisaTvis. SezRudva, rodesac ar gvaqvs masaTa Zalebi, araris arsebiTi, radgan kerZo amonaxsnis ageba, romelicSeesabameba am Zalebs, SesaZlebelia zogad SemTxvevaSi daadvilad sruldeba maTi kerZo mocemulobaTaTvis (wona,centridanuli Zalebi da a.S.). adreuli gamokvlevebis mimoxilvazogadi amonaxsnebis mixedviT mogvca p. papkoviCma (1937) [1]; maTiagebis erTiani xerxi, dafuZnebuli Zabvis funqciis tenzorisgamoyenebaze, SemogvTavaza i. krutkovma (1949) [2].zogadi amonaxsnebis agebis amocanaze yuradReba iqnamipyrobili 1930 wels gamoqveynebuli b. galiorkinis statiiT [3].naCvenebi iqna, rom drekadobis Teoriis gantolebebi ZabvebSi( Tˆ − ZabvaTa tenzori, σ - misi pirveli invarianti)∇T ˆ =20, (1 + v ) ∇ Tˆ+ ∇∇σ= 0 . (1)16

SeiZleba dakmayofildes, Tu gamovsaxavT gadaadgilebaTa uveqtors biharmoniuli G veqtoriT TanafardobiT22μ u = ∇∇ ⋅G− 2(1 − v)∇ G . (2)es amonaxsni im miTiTebiT, rom misgan miiRebian adrecnobili SemoTavazebuli iyo u. busineskis mier jer kidev 1889wels. p. papkoviCma (1937, 1939) miuTiTa, rom (2) warmoadgensgadaadgilebebSi drekadobis Teoriis zogad amonaxsns2(1 − 2v ) ∇ u + ∇∇ ⋅u= 0 . (3)es gamomdinareobs iqidan, rom (3) gantolebis struqtura,marjvena nawilSi masaTa Zalebis arsebobisas imeorebs (2)amonaxsnis struqturas. amitom, (2)-Si u veqtorad drekadobisTeoriis sasazRvro amocanis amoxsnis miRebisas (romelicakmayofilebs O tanis zedapirze sam pirobas), Cven SegviZliamovelodeT, rom G veqtoric SeiZleba daeqvemdebaros kidev sampirobas 0-ze, rac, rogorc Cans zedmetia.gadaadgilebaTa veqtoris povnisas harmoniuli veqtorisada χ skalaris gradientis jamis formiT2u − 4 (1 − v)B + ∇χ , ∇ B = 0 .(3)-Si Casmis Semdeg, mivdivarT gantolebamde∇ 2 χ = −2∇ ⋅ B .misi amonaxsni warmodgeba kerZo amonaxsnisaRniSnulia veqtor-radiusi). ase, romχ = −R ⋅ B jamiT (R-iTχ = −( R ⋅ B + B0), u = (1 − v)B − ∇ ⋅ ( R ⋅ B + B ) . (4)40drekadobis Teoriis gantolebis amonaxsnis aseTiwarmodgena mocemuli iqna p. papkoviCis mier (1932) da ramdenamdegvian g. neiberis [4] mier. p. papkoviCis gadmocemiT, igi ufro adrecnobili iyo g. grodskisaTvis. (4) gadaadgilebis veqtoriwarmodgenili harmoniuli veqtoris B da harmoniuli skalarisB0jamiT an oTxi harmoniuli funqciis B0, B s( s = 1,2,3)jamiT,sadacB s− B -s proeqciebia dekartes koordinatTa sistemisRerZebze. (4) amonaxsnis Caweris forma ekuTvnis i. arJanixsa [5]da m. slobodianskis (1954) da aqvs saxe17

u = 4 (1 − v)B + R ⋅∇B− R∇⋅ B . (5)igi gansxvavdeba (4)-gan (roca B0= 0 ) harmoniuli veqtorisagannulis toli divergenciiT (anu harmoniuli veqtoris rotoriT).SemoTavazebuli iyo agreTve amonaxsnTa formebi, gamosaxuliharmoniuli funqciebiT niutonis potencialebisagan moculobiTiintegralebis meSveobiT; aseTia ter-mkrtiCianis midgoma (1947)1 ∇ ⋅ Bu = 4 (1 − v)B + ∇∫dτ. (6)2π D RgadaadgilebaTa u veqtoris integraluri warmodgena misidivergenciisa da rotoris saSualebiT, agreTve sxeuliszedapirze u-sa da misi normaluri warmoebulis ∂ u ∂nmniSvnelobebi, mocemulia i. arJanixis mier.ramdenadac O sxeulis zedapirze gveZleva mxolod samisasazRvro piroba, miiCneva dasaSvebad (4) zogadi amonaxsnisgamosaxulebaSi SenarCundes mxolod sami harmoniuli funqcia,ukuvagdebT ra masSi, magaliTad B 0-s an erTerTs B sfunqciaTagan(g. neiberi). am sakiTxs ganixilavda p. papkoviCi (1939) da ufrodawvrilebiTYm. slobodianski (1954) [6].B0-is CarTva (4)-Si zedmetia, roca B0∇ veqtors warmovadgenTromeliRac B * harmoniuli veqtoriT, Tanafardobis meSveobiT**∇ B0 = 4(1− v)B − ∇R⋅ B . (7)magram maSin ∇ ⋅ B * = 0 , * *2∇XB = 0 , ise rom B = ∇θ, ∇ θ = 0 daTanafardoba (7) Caiwereba Semdegi saxiT∂θB = − − = ∑ ∞ 24(1 v)θ R R n Yn( θ,λ).∂Rn=0aq gamoyenebulia B 0harmoniuli funqciis warmodgena Sidacaladbmul midamoSi harmoniuli polinomebis mwkriviT R n Yn.maSin∑ ∞ nR Ynθ =, (8)n=0 4(1 − v)− n18

3da rodesac n=3, v=0,25, B0= R Y3(7) Tanafardoba ar SeiZleba⎛1 ⎞dakmayofildes ⎜ivaraudeba , rom 0 < v < ⎟ . aqedan, miiReba ra⎝2 ⎠mxedvelobaSi keldiS-lavrentievis Teorema sasrul caladbmulmidamoSi harmoniuli funqciis Tanabrad krebadi harmoniulipolinomebis mwkrivad warmodgenis Sesaxeb, unda davaskvnaT, rom(7) warmodgena, roca v=0,25-s, SeuZlebelia. usasrulomidamosaTvis RruTi B 0-is warmodgenaSi unda Seicvalos n − ( n +1)-iT, mniSvneli (8) mwkrivSi 4 (1 − v ) + n + 1 ar gadaiqceva nulad1arcerTi mTeli n-saTvis da 0 < v < ; (5) amonaxsni warmoadgens2zogads sasruli caladbmuli midamosaTvis, v=0,25 gamonaklisisgareSe, xolo usasrulo midamosaTvis RruTi v ≠ 0, 25 -isas. ufrozogadi amonaxsnebi SeiZleba moviZioT m. slobodianskisTan (1954).migvaCnia, rom (4) amonaxsnSi, iseve rogorc zogadiamonaxsnebis sxva formebSi, unda vxedavdeT drekadobis TeoriissasazRvro amocanebis amoxsnis sasargeblo damxmare saSualebas,romelic dauSvebs laplasis gantolebebis klasikuri kerZoamonaxsnebis uSualo gamoyenebas. konkretuli amocanis amoxsnisagebisas meoTxe harmoniuli funqciis SenarCuneba aadvilebs amamonaxsnTa arCevas (a.i. lurie, 1955) [7].v. bloxi (1958) [8] gamodis gadaadgilebaTa u veqtorisharmoniuli veqtorisa da χ skalaris gradientis jamiswarmodgenidan. (4)-is amonaxsni bloxis warmodgenaSi moipoveba2χ = −( R ⋅ B + B0)-isas miiCnevs ra, rom χ = −R∇ ⋅ C , sadac C _harmoniuli veqtoria, bloxi midis warmodgenamde2u = 2 (1 − v)R∇⋅C− 4(1 − v)R×( ∇× C)− R ∇∇ ⋅C, (9)romelic SeiZleba SevavsoT harmoniuli skalaris gradientiTada harmoniuli veqtoris rotoriT. bloxis warmodgenaSiSesulia kidev Sesakrebebi, romlebic gamosaxulia sami brtyeliharmoniuli funqciiT.19

amonaxsnTa formebi, miTiTebuli i. arJanixisa da f.Curikovis mier (1953)12μ u = (1 − 2v)B − ∇× ( R×B), (10)2ar gansxvavdeba arJanixisa da slobodianskis (5)-isagan.ufro zogadi forma aqvs miTiTebuli v. deevs (1959)[(4v− 3) β + 4(1 − v)(ε −δ)] B + ( ∇B⋅ +2μu = β ) R=2[ 2(4v− 3) −δ] R ⋅∇B+ εR∇∇ ⋅ Bε , (11)amasTan β, ε, δ _ mudmivebia, romlebic SeiZleba nebismieradavarCioT. am mudmivebis jerovani arCevisas vubrundebiT (4), (5),(9) amonaxsnebs. (11)-Si Sedis agreTve amonaxsni2(4v− 3) 122μ u = 4(1 − v)B + R∇ ⋅ B + R∇∇⋅ B,(12)7 −8v7 −8vromelic gamosaxulia B harmoniuli veqtoriT da misi ∇BdivergenciiT.o. liavis [9] cnobili amonaxsni aqsialuri simetriis (zRerZis garSemo) SemTxvevisaTvis gamomdinareobs (2)-dan TumiviRebT = χ( r,z), G = 0 , G = 0 . ufro zogadi warmodgenaG zxycilindrul koordinatebSi (harmoniuli da biharmoniulifunqciebiT) mocemulia s. gutmanis mier (1948) [10].mravalbmuli D0midamosaTvis, romelic SemosazRvruliagaredan O 0 zedapiriT da Signidan O i zedapirebiT ( i = 1, L,k),mTlianad mdebareobs D-Si da erTmaneTTan da O 0 -Tan saerTowertilebi ar gaaCnia, gadaadgilebis veqtori v ≠ 0,25-isaswarmovadginoT saxiT (m. slobodianski, 1959)uku = u( 0)+ ∑u( i),B = B0 + ∑ B( i),i=1= 4 (1 − v)B(0)− ∇R⋅(0),, u( i)= 4(1− v)Bi− ∇R(i)⋅ B(i),0Bki=1sadacxoloB (i)− veqtoria, harmoniul midamoSi, gareTa O i -is mimarT,B − harmoniuli D − Si, amasTan saTave Ω i R i veqtorisa(0)0ganlagebulia RruSi, romelic SemosazRvrulia O i . am amonaxsnis20

forma _ `srulia~, Tu sxivi Ω i -dan kveTs O i -s erT wertilSi; isiqneba `saerTo~, roca R i -liapunovis Caketili zedapiria.aseve unda aRiniSnos, rom wrfivi diferencialurigantolebebis sistemis `zogadi amoxsnebis~ agebis amocanan∑j=0L = 0 ( i = 1,2, L,n)iju j(masSiLij_wrfivi diferencialuri operatorebia mudmivikoeficientebiTx , x2,, x1LmcvladebiT) daiyvaneba (a. lurie, 1937;ufro mogvianebiT (1953 w.) rumineli mecnieri g. moizeli)`potencialebis~ ϕs( s = 1, L,n)moZebnamde, romlebiTac u jamonaxsnigamoisaxeba Tanafardobis meSveobiT, saxiTu=n∑jM sjs=1ϕs( j = 1,2, L,n)sadacMsj− saZiebeli wrfivi diferencialuri operatorebia, xoloTiToeuli _ ϕspotencialTagan, akmayofilebs erTi da imavediferencialur gantolebasKϕ = 0 . ( s = 1,2, L,n)sadvilad SeiniSneba, rom operatori K = LijwarmoadgensoperatorebisLijkvadratuli matricis ganmsazRvrels, xoloMsj_ am ganmsazRvrelis j-uri svetis algebrul damatebas.gadaadgilebebSi drekadobis Teoriis gantolebebisadmimoxmarebisas, izotropuli tanebisaTvis, aRweril gamoTvlasmivyavarT galiorkin-busineskis (2) amoxsnamde. aSkaraa, rom esxerxi, gamodgeba anizotropuli garemosaTvis, drekadobisTeoriis dinamikuri gantolebebisaTvis da a.S.1.1.2. ZabvaTa funqciis tenzoriviciT, rom rotors Φˆ tenzoris transponirebuli rotorisaewodeba araTavsebadobis ( I ik) tenzori Φˆ -ze:21

I ˆ ˆ )TnkΦ = ∇× ( ∇× Φ . (13)es – simetriuli tenzoria, Tu Φˆ tenzori simetriulia. InkΦˆ-issxvagvar warmodgenas aqvs saxe2 2I Φ = −∇ Φ + ∇ ⋅Φ − ˆ∇ ⋅∇ ⋅Φ( ˆnkˆ ˆ 2det ˆ E E∇− ∇∇)Φ, (14)aqÊ − erTeulovani tenzoria, Φ = I Φˆ) − Φˆ− is pirveli invarianti;12defa = ( ∇a+ ∇a)− operacia a veqtorze, romelsac uwodeben am2veqtoris `deformacias~. magaliTisaTvis gamodgeba deformaciiswrfivi tenzori εˆ = defu . uwyveti (sen-venanis) pirobebigamoxataven am tenzoris nulad gadaqcevas:1 (Inkε ˆ = I defu = 0 . (15)nksazogadod, yoveli veqtorisaTvis I nkdefa = 0 . piriqiT, TuI ˆnkΦ = 0 , maSin Φˆ= defa − arsebobs veqtori, romlis deformaciebiwarmoadgens Φˆ tenzors.kerZod, tenzorisaTvis E ˆΦ= EI ˆ ,( Φˆ)iˆΦ = ˆ 2I nkE ( E∇− ∇∇)Φ , (16)xolo nulis toli divergenciis tenzorisaTvis ( ∇ ⋅Φ = 0)InkΦˆ 2= −∇ Φ ˆ + InkEˆΦ . (17)moculobiTi Zalebis ar arsebobisas aseT tenzorswarmoadgens ZabvaTa Tˆ tenzori; Tu SemovitanT σ = I ( ˆ1T )aRniSvnas, (16) da (17) Tanaxmad gveqneba( ˆ∇− ∇∇)σ)ˆ 2 ˆ 2I nkT = −∇ T + E , ( ∇ ⋅T ˆ = 0). (18)Tu mivmarTavT axla hukis kanons izotropuli tanisaTvisv2μεˆ= Tˆ− σEˆ, (19)1+v(15) da (17)-is Tanaxmad gveqneba( Eˆ∇ − ∇∇) σ = 0,2 ˆ 1 22− ∇ T +∇ σ = 0 . (20)1+vamasTan meore Tanafardoba miRebulia tenzoris pirveliinvariantis warmoSobiT (20)-is marcxena nawilSi. amrigad,miRebulia beltrami-mitCelis damokidebulebebi22

2 2 ∇∇σ∇ T + = 0 . (21)1+vcnobilia, rom tenzori nulis toli divergenciiTgamosaxvadia meore tenzoris rotoriT: _ Tu ∇ ⋅Tˆ = 0 , maSinTˆ ∇ × CˆT= ; Tu, amasTanave tenzori Tˆ simetriulia ( Tˆ = Tˆ), maSinsimetriuli Φˆ tenzoris ganxilvaSi Semotanisas, gvmarTebsmiviRoTe ˆ ˆ )ˆ = ˆ ˆTT= ( ∇× Φ ; maSin ∇ × ( ∇ × Φ)= InkΦˆ Tˆ ˆ =nknk= ˆ ,TT , T ( I Φ) = I Φ Tradganac pirobis mixedviT Φˆ T = Φˆ. aqedan gamomdinareobs, romuwyveti garemos statikis gantolebebi, masaTa Zalebis ararsebobisas ˆ 0, ˆ ˆ T( ∇ ⋅T = T = T ) kmayofildeba, Tu miviRebTT ˆ = Φˆ, ˆ ˆ T( Φ = Φ ) . (22)I nki. krutkovis (1949), v. bloxis (1964) da b. fincis mierSemotanil Φˆ tenzors uwodeben ZabvaTa funqciis tenzors.ZabvaTa Tˆ tenzori rCeba ucvleli, TuΦˆ -is gamosaxulebaSiSemotanilia Sesakrebi saxiT defa, sadac a – nebismieri veqtoria.es iZleva Φˆ tenzoris mocemis formis gamartivebis saSualebas,SevinarCunebT ra mis gamosaxulebaSi mxolod sam komponents.aseTia maqsvelis (tenzori Φˆ − diagonaluria) da moreris ( Φˆ − SiSenarCunebulia mxolod aradiagonaluri komponentebi)warmodgenebi.v. bloxis wignSi (1964) miTiTebulia dekartulkoordinatebSi Φˆ -is kidev sami komponenturi forma; iqveCamoTvlili cilindrul koordinatebSiΦˆ − is samkomponenturiwarmodgenis cxra varianti brunvis simetriis SemTxvevisaTvis (i.krutkovi, 1949, gv. 108).i. krutkovis gardaqmna. vubrundebiT ra drekad izotropulgaremos da viTvaliswinebT ra, rom (20) da (16)-is mixedviT∇∇ σ = −I ˆnkEσ , SeiZleba warmovadginoT beltrami-mitCelisdamokidebulebebis saxiTI nk⎛ ˆ σ ˆ ⎞⎜Φ− E ⎟ = 0 . (23)⎝ 1−v ⎠23

ase, rom frCxilebSi tenzori warmoadgens deformacias,romeliRac C veqtorze∇ Φˆ− Eˆ= defc . (24)1+vamasTan erTad (22)-is Tanaxmad22σ = I ( InkΦˆ ) = ∇ Φ − ∇ ⋅ ∇ ⋅ Φ ˆ = ∇ Φ − ∇ ⋅ b , b = ∇ ⋅ Φˆ, (25)12 σes iZleva saSualebas (24) CavweroT saxiTEˆ2∇ Φˆ 2− ( ∇ Φ − ∇ ⋅b) = defc . (26)1+vam Tanafardobidan gamoiricxeba veqtori c; tenzorzepirveli invariantebis SeqmniT (26)-Si, mivdivarT tolobamde⎛ 3 2 − v ⎞∇ ⋅⎜v − ∇Φ − c⎟= 0 ,⎝1+v 1+v ⎠romelic gamosaxavs frCxilebSi veqtoris divergenciis nuladgadaqcevas; es veqtori warmoadgens sxva veqtoris rotors,magram es ukanaskneli SeiZleba CavrToT b veqtorisSemadgenlobaSi. amiT ganisazRvreba c da mere defc; (25)-Si Casmas,mivyavarT gantolebamdeTu2∇ Φˆ1 23 2 − v( ∇ Φ − ∇ ⋅b) + defb − ∇∇Φ2∇ Φ = Eˆ1+v 1+v 1−v. (27)− s gamovricxavT (14), (22) da (27) gamosaxulebebidan,mivalT izotropul drekad garemoSi ZabvaTa Tˆ tenzoriswarmodgenamde:1−2v( ∇ Φ − ∇ ⋅b) − ( − ∇∇Φ)vTˆ = Eˆ2defb . (28)1+v1+vaxla (19)-is Tanaxmad ganisazRvreba deformaciis εˆ tenzorida masze dayrdobiT gadaadgilebis u veqtori (myari tanisgadaadgileba ukuigdeba):1−2v1−2v2μ ˆ ε = def ( ∇Φ − b), 2μ u = ∇Φ − b . (29)1 + v1 + vi. krutkovis (28) da (29) formulebi warmoadgenendrekadobis wrfivi Teoriis gantolebebis zogad amonaxsnTa erTformaTagans; maTi meSveobiT ganisazRvreba ZabvaTa Φˆ funqciis24

tenzori, romelic akmayofilebs (27) diferencialur gantolebas,ZabvaTa T tenzori da gadaadgilebaTa u veqtori. es ukanasknelniaRmoCndnen mxolod I 1( Φ) ˆ = Φ da b = ∇ ⋅Φˆ-ze damokidebulebebi.amitom sakmarisia (27)-is saSualebiT, davadginoT Tanafardobamxolod am sidideebs Soris. Sesabamisad, Tu SevadgenTdivergencias (27)-is marjvena da marcxena nawilebSi, miviRebT2 ∇∇ ⋅b2(1 − v)2∇ b + = ∇∇ Φ . (30)1−2v1−2vigulisxmeva, rom∇Gb = ∇ 2 G, Φ = . (31)2(1− v)Cven davakmayofilebT am (30) gantolebas, Tu veqtori c –biharmoniulia. (29)-is Tanaxmad amas mivyavarT, galiorkinbusineskis(2) amonaxnamde.(30) gantolebis kerZo amonaxsnad gamodgeba b = ∇Φ, xoloSesabamisi erTgvarovani gantoleba (nulis toli marjvenanawilia) mxolod mudmivebis mniSvnelobebiT gansxvavdeba (1.3)gantolebisagan gadaadgilebaTa veqtorisaTvis. amitom b veqtoriSeiZleba agebuli iqnas papkoviCis amoxsniT (4) tipis mixedviT:1+vb = [ 4(1 − v)B − ∇(R ⋅ B + B 0)] + ∇Φ(32)1−2vda (1.29)-Si Casmas mivyavarT u veqtoris (4)-iT warmodgenamde.(30), (28) da (29) formulebze dayrdnobiT a. krutkovma miiRomravalricxovani sxva `zogadi amonaxsnebi~. magaliTad, TuSemovitanT ganxilvaSi K veqtors nulis toli divergenciiT darotoriT, romelic ganisazRvreba b veqtoriTa da ∇Φ-iT:b∇ ⋅ K = 0 , ∇ × K = ∇Φ − ,2(1− v)maSin (29) da (30) Tanaxmad, ukuvagdebT ra ara arsebiT mudmivTanamamravls, mivalT kornis amoxsnamde2( ∇ K = ∇× b,∇ ⋅ = 0)u = ∇× K − (1 − 2v)bK .25

1.1.2.1. sivrciTi amocanis integraluri gantolebebisivrciTi sasazRvro amocanebis integraluri gantolebebisSedgena, maT SeswavlasTan dakavSirebuli siZneleTa gadalaxva,arsebobis damtkiceba da maTi amoxsnebis agebis efeqturixerxebis povna _ v. kupraZis da misi TanamSromlebis mravalwlianimuSaobis Sedegia. am gamokvlevebis meTodebisa da SedegebisSinaarsis gadmocema, dawvrilebiTi bibliografiiT, moipovebaagreTve v. kupraZis, T. gegelias, o. baSaleiSvilis daT.burWulaZis monografiaSi, romelic gamoqveynda 1968 wels [11].am naSromSi ganvixilavT sivrciTi drekadobis Teoriismxolod pirveli da meore sasazRvro amocanebi izotropulierTgvarovani garemosaTvis. Cven amasTan SemovifarglebiTcalbmuli sasruli moculobisaTvis( V i) − Sida (i) amocaniT dagare (e)-Ti. RruTi aRWurvili usasrulo garemosTvis ( V e)ivaraudeba O zedapiris sigluve, romelic sazRvravs Vi-sgaredan ( V − DSignidan).edrekadobis Teoriis potencialebi. ganvixilavT Semodisu ˆ(M , Q)kelvin-somilianis tenzori, romelic gansazRvravsSemousazRvreli drekadi garemos M wertilis u ( M , Q)gadaadgilebas, gamowveuls Q wertilSi erTeulovani SeyursuliZalis moqmedebiT:ˆ⎡ ˆ ⎤u( M , Q)= U ( M , Q)⋅e,ˆ 1 E ∇∇KU ( M , Q)= ⎢ − ⎥ , (33)4πμ⎣ R 4(1 − v)⎦( Ê − erTeulovani tenzoria, R = QM = ΓM− ΓQ, R = R ), Zabvis veqtorisgamosaxuleban ˆ ˆμ⋅T= Φ(M , Q)e,Φˆ1( M , Q)=[(1− 2v)( nμR − Rnμ) −38π(1 − v)R⎤− − v En ˆ 3 12(1 )μR⋅R nμ⋅ R∇∇R ⎥. (34)⎦vTqvaT O _ Caketili zedapiria ( M ⊂ 0); maSin26

∫∫ ×Φ(M , Q)dO μ=0R ˆ0(35)da adgili aqvs gausis ganzogadebul Teoremas∫∫0Φˆ( M.Q)dO= −Eˆδ ( Q)μ,⎧ 1 Q ⊂ Vi⎪δ ( Q)= ⎨12 Q ⊂ O(36)⎪⎩ 0 Q ⊂ Vt( V i− moculoba O-s SigniT, Ve− O-s gareT).v. kupraZis mier Semotanili drekadobis Teoriis veqtorebipotencialebidan ganixileba ori: pirveli, msgavsi O zedapirze.martivi fenis A (Q)potenciali da meore msgavsi ormagi fenisB (Q) potencialiA ( Q)= ∫∫a(M ) ⋅Uˆ ( M , Q)doμ, (37)0B Q)= b(M ) ⋅Φˆ( M , Q)do . (38)∫∫aSkaraa, rom A(Q) da B(Q),(μ0Q ⊄ 0 − isas, warmoadgenen drekadobisTeoriis gantolebebis amonaxsns gadaadgilebebSi, rodesac argvaqvs moculobiTi Zalebi.pirveli potencialis zRvruli mniSvnelobebi O-zeSignidan da garedan, romlebic aRniSnuliaA ( Qi) = lim ( A(), A ( Q ) = lim ( A() − Ti0QVi⊃Q→Q0e0QVi⊃Q→Q0toli arian misi pirdapiri mniSvnelobis, gansazRvrulisarasakuTrivi krebadi integraliTA Q0 ) = ∫∫a(M )0⋅Uˆ ( M , Q0) do . (39)(μ0meore potencialis zRvruli mniSvnelobebisaTvis adgiliaqvs Tanafardobebs11B i( Q0 ) = B(Q0) − b(Q0), B e( Q0 ) = B(Q0) + b(Q0) , (40)22analogiurebs plemelis formulebisa, amasTan pirdapirimniSvneloba ganisazRvreba integraliT, romelic krebadia mxolodmTavari mniSvnelobis azriTB ( Q ) = b(M ) − Φˆ( M , Q ) do = lim b(M ) ⋅Φˆ( M , Q ) do ,0∫∫00μ∫∫ε →0o−o(Q0ε)0μ27

R −r Q( O( Q0 , ) − Q0ε wertilis O-ze midamoa, 2ε diametriT).Q ⊂ V ewertilis sakmarisi moSorebiT O zedapiridan,→ Tanaxmad (33) da (37)-isa gvaqvs ( e = ν )1lim A(Q)=→Q∞16πμ(1− v)ΓQQQr Q[(3− 4v)Eˆ− e e ] a(M ) do ;es – Q-Si gadaadgilebis veqtoria, koordinatTa saTaveSimodebuli Zalis moqmedebiT, romelic moicema integraliT O-zesimkvrivisagan a(M).QQQ∫∫aμmeore potencialiQ → Q∞isas ufro nela ara xdeba nuli,vidre−2Γ ada is SeiZleba ganimartos rogorc gadaadgileba,romelic iqmneba ZalTa sistemiT, ganawilebuliT O zedapirze,nulis toli mTavari veqtoriT.integraluri gantolebebi. pirvel sasazRvro amocanaSiu(Q) gadaadgileba, romelic iRebs O zedapirze ( V imoculobaSida amocanaSi, Rru gare amocanaSi) mocemul mniSvnelobas,iZebneba drekadobis Teoriis meore potencialis formiT, ucnobisimkvriviT b(M):u Q)= B(Q)= b(M ) ⋅Φˆ( M , Q)do .(41)∫∫(μ0gare amocanis SemTxvevaSi es warmodgena miiCnevs, romu( Q ) ∞-s aqvs r 2 rigi; im ZalTa mTavari veqtori, romlebic unda− Qiyvnen ganawilebuli Rrus O zedapirze, rom mianiWon maTwertilebs u Q ) gadaadgilebis veqtori, toli unda iyos nulis.(0amitom pirveli sasazRvro amocanis amonaxsni (41) formiTSeiZleba arsebobdes mxolod u Q ) specialuri amocaniT, xolo(0zogad SemTxvevaSi amonaxsni iqneba warmodgenili (41) jamiT daubralo fenis potencialiT (`robenis elastostatikuriamocanis~ amonaxsni).Sida (i) da gare (e) amocanebis integraluri gantolebebimiiReba (41)-dan, zRvruli gadasvlis lim u(Q)= v(Q0)plemelis (4) formulebis daxmarebiT:Q→Q0gziT,28

( )I i( )I e1L b(Q0) − ∫∫b(M ) ⋅Φˆ( M , Q0) doμ= −v(Q0),(42)201L b(Q0) + ∫∫b(M ) ⋅Φˆ( M , Q0) doμ= −υ( Q0) . (43)201(30) da (36)-ze dayrdnobiT, roca δ ( Q)= , ar aris Zneli2davrwmundebiT, rom b(M)-is mocema myari tanis gadaadgilebisformiTxb ( M ) = + ω × r = υ + ω ×Γ + ω × R(44)υ0μ 0Q0warmoadgens (43)-is Sesabamisi erTgvarovani gantolebis amonaxsns( ) 1I e 0L b(Q0) = ∫∫ b(M ) ⋅Φˆ( M , Q0) do μ= 0 . (45)20amasTan erTad B( M ) = −b(M ) warmoadgens (42) gantolebisamonaxsns, roca O zedapiri gadaadgildeba rogorc myari tani,maSin mTeliVimoculoba aseve gadaadgildeba rogorc myaritani; rac gamomdinareobs (41) da (36)-dan, roca _ δ ( Q)= 1.meore sasazRvro amocanaSi O-ze moicema zedapiruli ZalebiF = ( n ⋅Tˆ) 0, xolo gadaadgilebaTa veqtori iZebneba pirvelipotencialis formiTu ( q)= ∫∫a(M ) ⋅Uˆ ( M , Q)doM. (46)0(34)-is gamoyenebiT, aq gamotovebuli (magram aratrivialuri)gardaqmnebis Semdeg, mivdivarT integralur gantolebamdeamasTannII( i)1L a(Q0) + ∫∫Φˆ ( Q0, M ) ⋅ a(M ) doM= F(Q0) = ( n0⋅Tˆ)0,(47)20( ) 1IIe L a(Q0) − ∫∫Φˆ ( Q0, M ) ⋅a(M ) doM= −F(Q0) = ( n0⋅Tˆ)0. (48)2a−Vi0− isadmi gare normalia.zemoT vaCveneT, rom (42)-(43), (47)-(48) veqtorul gantolebebSiintegralebi ganixileba maTi mTavari mniSvnelobebis azriT –gantolebaTa es sistema singularulia. Semdgomi gamokvlevebissirTule mdgomareobs maT mimarT Teoremisa da fredholmisalternativis miyenebis damtkicebaSi ( μ da v-isas, romlebic29

uzrunvelyofen deformaciis potenciuri energiis dadebiTobas);ix. v. kupraZe (1963, 1968) da agreTve s. mixlini (1962) [12].gadavweroT miRebuli gantolebebi aseTi TanmimdevrobiT:I( i)III1L b(Q2( e)( e)II( i)01L a(Q21L b(Q2) −001L a(Q2( i)( e)( e)( i)( I , II ) da ( , II )∫∫0) −) +0) −b(M ) ⋅Φˆ( M , Q)do∫∫0∫∫0∫∫0Φˆ( Q , M ) ⋅a(M ) do0Mb(M ) ⋅Φˆ( M , Q ) doΦˆ( Q , M ) ⋅a(M ) do00= υ(Q )MMM0= −F(Q0⎫⎪⎬ , (49)) ⎪⎪⎭⎫= υ0( Q0)⎪⎬ . (50)= F(Q ⎪0)⎪⎭I warmoadgenen mokavSire wyvilebs.Sesabamisi erTgvarovani gantolebebi Caiwereba Semdegnairad:( i)( e)1I0, I L b(Q0) − λ∫∫ b(M ) ⋅Φˆ( M , Q0) doM= 0,(51)20( i)( e)1II0 , II L a(Q0) − λ∫∫ Φˆ( Q0, M ) ⋅a(M ) doM= 0.(52)2( i)eamasTan λ = 1 I II) − Tvis da λ = −1( e ) , ( iI II) − Tvis.00,( 00(i)I da(e)II amocanebis amoxsnis arseboba da erTaderToba.sakmarisia SevamowmoT, rom λ = 1 ar warmoadgens(e)II erTgvarovanigantolebis sakuTar ricxvs (maSasadame, misi mokavSireI( i)0gantolebisac). mtkicdeba, rom daSveba II (e) − is amonaxsnisarsebobis Sesaxeb, romelic gansxvavdeba trivialurisagan( a ( M ) ≠ 0), SeuTavsebelia deformaciis xvedriTi potencialurienergiis dadebiTobis moTxovnasTan. fredholmis TeoremisTanaxmad aqedan gamomdinareobs(e)II da(i)I araerTgvarovanigantolebebis amonaxsnebis arseboba da erTaderToba F Q ) -isnebismierad mocemisas pirvelSi da υ(Q)-is meoreSi.I( e)0(i)II meore Sida sasazRvro amoxsna. II(i) −(0sTan mokavSire-s aqvs aratrivialuri amonaxsni (44). amitom aratrivialuriamonaxsniaqvsII -sac, da fredholmis erTerTi Teoremis( i)030

(i)Tanaxmad araerTgvarovan gantolebas II SeiZleba qondesamonaxsni, mxolod im pirobis gaTvaliswinebiT, rom misiTavisufali wevri (44)-is orTogonaluria:υ0∫∫0+ ∫∫ ∫∫( υ ω × r ) ⋅ F(Q)do = υ F(Q)do + ω r × F(Q)do 0 .0 QQ 0Q QQ=00− isa da ω-s nebismierobis gamo amas mivyavarT zedapiruliZalebis mTavari momentisa da mTavari veqtoris nulad qcevispirobebamde. maTi Sesrulebisas(i)II amocanis amoxsna gansazRvrulia( e)myari tanis gadaadgilebis Sesakrebis _ I mokavSire gantolebissakuTar amonaxsnis sizustemde.01.1.2.2. robenis elastostatikuri amocanarobenis elastostatikur amocanas uwodeben potencialisgansazRvras velSi, romelic gars artyia Caketil gamtarzedapirs masze mocemuli muxtiT. drekadobis TeoriaSi terminiSemoitana v. kupraZem (1963) [13] iZebneba daZabuli mdgomareobaSemousazRvrav drekad garemoSi, roca masSi CarCilul myarsxeuls eniWeba gadaadgilebaω0 )0( Q = u + ω × Γ . (53)amocanis amonaxsni iZebneba martivi fenis potencialisformiTQ0ω ( Q ) a ( M ) Uˆ ( M , Q)do M, (54)= ∫∫0da mtkicdeba, rom simkvrivis a 0 ( M ) veqtorad am SemTxvevaSi undamiviRoT(i)0II amocanis sakuTari amonaxsni, amasTan _ F ( Q0 ) = a ( Q0)(es uSualod gamomdinareobs(e)II integraluri gantolebidan).amrigad, _ a 0 Q ) gansazRvravs O gadaadgilebuli myari tanis(0zedapirze reaqciuli garemos masze ganawilebas. vuwodoT a k Q )da a k+ 3 Q ) O-ze am Zalebis ganawilebas, romelic tanze(0(0erTeulovani ZaliskV = ik+3k−s, romelic modebulia i kRerZze da31

Sesabamisi erTeuli momentisk+3m = ir+3k, moqmedebiTaa gamowveuli.vTqvaT, Semdegru = i ,aseTi aRniSvnebisasrr+3U = i × ω − I sakuTar amonasxnTa sistemaa.r( e)0k∫∫ ⋅ u doQ=0ra δ ( k , r = 1,2, L,6). (55)kramiT ganisazRvrebaII integraluri gantolebis sakuTar( i)0amonaxsnTa sistema, orTonormirebuliI0e( )− sakuTari amonaxsnebissistemasTan.(e)pirveli gare sasazRvro amocana. ( I ) amocanas aqvsamonaxsni, Tu Tavisufali wevri I (e)gantolebisa orTogonaluriaII amocanis a 0 ( M 0 ) sakuTari amonaxsnisa( i)00∫∫ ( Q ) a ( Q0) doQ=0υ . (56)00es piroba amocanis arsiT ki ar aris gamowveuli, aramed u(Q)-ismiRebuli warmodgeniT meore potencialis formaSi. ganxilvaSiSemodis veqtoridar6∑*υ ( Q ) = υ(Q ) − D ru(57)00D mudmivebi ganisazRvreba ise, rom (56) orTogonalobispiroba sruldebodes am veqtorisaTvis. (55)-is Tanaxmad gvaqvsr=1*kk∫∫ ( Q0) − a ~ Q0) doQ∫∫υ(Q0) ⋅ a ( Q0) do − =0 QD0 k0υ 0 ,Tu axla miviCnevT0rmiviRebTu0=ω =03∑k = 13∑vi∫∫k0∫∫kk = 1 0k⎫υ(Q0) ⋅ a(Q0) doQ,0⎪+ 3⎬ . (58)kυ(Q ⋅ ⎪0) a ( Q0) doQ0⎪⎭( u + r )*( Q 0)= υ ( Q 0)− ω × 0Q0υ . (59)(e)I amocanas, υ* ( Q0)-is toli Tavisufali wevrisas, aqvsamonaxsni da misiT ganisazRvreba Q ⊂ V -isas veqtori u * ( Q ) .robenis amocana (54) formaSi ixsneba gansazRvruli (58) Tanaxmad32

gadaadgilebisaTviswarmoadgens jamsu0+ ω × rQ 0da I (e)amocanis saZiebeli amonaxsni*u Q)= u ( Q)+ ω ( Q), Q ⊂ V ).(60)(0(ev. kupraZis wignebSi (1963, 1964, 1967, 1968) ganxiluliaintegraluri gantolebebi da maTi amonaxsnebis arsebobissakiTxebi ara marto statikis amocanebisaTvis, aramed drekadigaremos damyarebuli rxevebisTvisac. ganxilulia anizotropulida araerTgvarovani garemoebebi, rigi sxva sasazRvro amocanebi,daTmobili aqvs adgili Termodrekadobis amocanebs, agreTveamocanebs SemosazRvruli moculobebisa da usasrulogaremosaTvis ramdenime darTuli RruTi. daZleulia Sesaswavliintegraluri gantolebebis singularobasTan dakavSirebulisiZneleTa rigi. SemoTavazebulia ideaSi (magram ararealizaciisas) am gantolebaTa ricxviTi amoxsnebis martivixerxebi.(i)I amocanis integralur gantolebas 1907 wels ganixilavdaj.lauriCela; d. Sermanma (1962) ganazogada amonaxsni sasrulimoculobis drekadi tanis SemTxvevisaTvis ramdenime RruTi.amoxsnis miaxloebiTi meTodebi, dafuZnebuli variaciuliprincipebis gamoyenebaze. ganviTarebulia l.leibenzonis (1951) [14]monografiaSi. variaciuli meTodebis krebadobis Seswavlisa dacdomilebaTa Sefasebisadmia miZRvnili bevri gamokvleva (s.mixlini [15], m. slobodianski).1.1.3. drekadobis Teoriis sivrciTi amocanebisSesaxebdrekadobis Teoriis sivrciTi amocanebis sistematuriSeswavliT dakavda b. galiorkini. iyenebda ra sami biharmoniulifunqciiT drekadobis Teoriis gantolebebis zogadi integraliswarmodgenas (1930) da mwkrivebs, is anviTarebda ocdaaTianiwlebis dasawyisidan sqeli filebis angariSis meTods, romelicgulisxmobda pirobebis Sesrulebas nebismieri datvirTvebisaTvis33

torsebze da integraluri pirobebisa gverdiT zedapirze; manSeiswavla marTkuTxa, mrgvali, seqtoruli, samkuTxa filebi (1931,1932). 1931-Si galiorkinma aago amocanis amoxsna normaluridatvirTvis moqmedebis qveS moqceuli fenis wonasworobisSesaxeb mwkrivebis daxmarebiT, romlebic Seicaven beselisa dahankelis funqciebs. galiorkinma ganixila amocana Rru cilindrisada misi nawilis wonasworobis Sesaxeb (1933), xolo mogvianebiTmiiRo Rru sferos RerZulsimetriuli deformaciis amocaniskerZo amonaxsnebi (1942).am gamokvlevaTa kvaldakval gamoCnda g. maslovis (1938)naSromi, romelSic ganixileboda Termodrekadi wonasworobasqeli filis, Rru cilindrisa da sferos stacionaruliTermuli nakadis moqmedebisas.busineskis amocanis ganzogadeba naxevarsivrcisTvismocemulia v.korotkovis mier (1938), romelmac gamoikvliamarTkuTxedze _ mudmivi da wrfivi kanoniT cvladi datvirTvismodebis SemTxveva. naxevarsivrcis amocanebi mocemulobisaTvisgadanacvlebis sazRvarze da erTmaneTTan SeuRlebulinaxevarsivrceebis SemTxveva ganixileboda d. Sermanis (1943, 1945)mier. `centr~-is tipis Taviseburebis mqone amonaxsni,naxevarsivrcis romeliRac wertilSi, miRebulia v. fedianinismier (1965).ormocdaaTiani wlebis meore naxevridan gamoCnda naSromebi,romlebic ganixilaven naxevarsivrcis (n. rostovcevi, 1955 [16];b.mincbergi, 1957) da drekadi fenis grexas (i. ufliandi, 1959).mravalfeniani garemos (fuZis) grexis SemTxveva ganxilulia v.petriSinis mier (1965); orfeniani garemos grexa Seswavlilia d.grilickis mier (1961) [17].busineskis tipis amocanebi anizotropuli garemosaTvisganixila c.sveklom (1964). Cndeba gamokvlevebi, romlebicSeiswavlian araerTgvarovani garemosagan naxevarsivrcis qcevas:s. lexnickim (1962) gamoikvlia naxevarsibrtye da solidrekadobis cvladi moduliT, l. ter-mkrtiCianma (1961) ganixila34

sivrciTi amocanebi araerTgvarovani garemosaTvis (busineskisamocana, simetriulad datvirTuli cilindri). araerTgvarovaninaxevarsivrcisa da naxevarsibrtyis ufro zogadi saxeSeiswavleboda n.rostovcevis mier (1964); busineskis amocanawrfivi deformadi mTliani garemos specialuri tipisaTvisdaisva da amoixsna a. vinogradovis mier (1966) [18]. Termodrekadiamocana naxevarsivrcisaTvis, romelic esazRvreba garemos daromlis temperaturac gveZleva ganawilebis gausis kanoniT,ganxilulia i. kilis mier (1966).drekadi fenis wonasworoba furies integralis meSveobiTSeiswavleboda g. Sapiros mier (1942, 1944); man Seiswavla amocanawnevis gadacemis Sesaxeb farTobze ganawilebuli wrisaTvis; d.aizenbergTan (1950) TanaavtorobiT mis mier ganxilul iqna wnevisgadacema wriuli xvrelis mqone fenidan. wnevis gadacema feniT,romelic devs drekad fuZeze, fenasa da fuZis sruliSerwymis pirobisas Seswavlilia r. rapoportis mier (1948).sqeli filis Runva zedapiruli harmoniuli datvirTvismoqmedebisas gamoikvlia s. gutmanma (1940); mis miervea amoxsnilisqeli filis sakuTari woniT Runvis Sesaxeb amocanas (1941);ufro mogvianebiT sqeli filebis Runvis sakiTxebiT bevriavtori iyo dakavebuli (a. aleqseevi, 1946; v. bloxi, 1954, m. huseinzade,1956; v. prokopovi, 1963).1942 wels a. luriem SemogvTavaza drekadi fenisa da sqelifilis wonasworobis Sesaxeb amocanis amoxsnis axali simbolurimeTodi, dafuZnebuli drekadobis Teoriis sivrciTi amocanisgantolebebis amoxsnis warmodgenaze laplasis organzomilebianioperatoris mTeli transcendenturi funqciebis saxiT. aseTmawarmodgenam saSualeba mogvca gagvemartivebina kompaqturadCaweril xarisxovan mwkrivebze simboluri operatorebismeSveobiT moqmedebebi da amis garda bunebrivi saxiT migviyvanaaxali klasis amoxsnebis moZebnamde, romlebic filis gverdiTzedapirze sasazRvro pirobebis Sesrulebis dazustebissaSualebas iZlevian. am amonaxsnebs luriem `erTgvarovani~35

uwoda, radganac isini filis torsebze datvirTvis ar arsebobispirobas Seesabamebian.luries simboluri meTodi SemdgomSi filebis TeoriissferoSi gamoiyenes e. krugim (1956), i. teregulovma (1961), t.xaCaturianma (1963) da u. nikulinma (1963). v. agariovis (1963) [19]monografiaSi filebis Teoriisadmi simboluri meTodisgamoyenebis sfero farTovdeba. filebis Teoriisadmi simbolurimeTodis Semdgomi miyeneba potencialuri energiis minimumisprincipTan SerwymiT mocemulia v. prokopovis mier (1965).s.lexnickis naSromebSi simboluri meTodi gamoiyenebatransversalur-izotropuli fenisa da sqeli filiswonasworobis ganxilvisas; mis mier miRebulia agreTveSesabamisi erTgvarovani amonaxsnebi. p. nedorezovma (1964) amoxsnasimboluri meTodiT amocana mravalfeniani Rru cilindrisgrexis Sesaxeb.amonaxsnebis simboluri warmodgenis daxmarebiT a. luriem(1955) advilad daadgina, rom SemousazRvrel filaSi ( z < h)nulisagan gansxvavdeba Tburi Zabvis tenzoris mxolodσ , ,xσyτ xykomponentebi. isini gamoisaxebian M(x, y, z) funqciiT, romelic aqTamaSobs brtyel amocanaSi r-is funqciis msgavs rols. funqciaM ganisazRvreba kvadraturebiT temperaturis mocemulistacionaruli ganawilebis kanonis mixedviT.furies gardaqmnebis daxmarebiT s. dimkovma (1966) amoxsnaamocana drekadi fenis wonasworobis Sesaxeb. aseTma midgomamsaSualeba misca avtors mieRo aseve asimptoturi formulebiamoxsnisaTvis.d. martinenkom (1964) miiRo amonaxsni mwkrivebSi fenisaTvis,mis sazRvrebze mocemuli gadaadgilebebisas (meore ZiriTadiamocana). fenis Seyursuli Zalis moqmedeba ganixiles o. Sextermada o. prixodCenkom (1964); maT mier miRebul iqna amonaxsnivertikaluri Zalis moqmedebis Sesaxeb fenis SigniT, romelicdevs kldovan fuZeze. cvladi sisqis fenisa da cvladi sisqis36

mrgvali filis SemTxveva datvirTvisas, romelsac gaaCniaRerZuli simetria garCeulia i. semionovas mier (1965).mrgvali sqeli filis wonasworoba, romelzec moqmedebsTanabrad ganawilebuli datvirTva, Seswavlili iqnaerTgvarovani amonaxsnebis saSualebiT g. buxarinovis mier (1952),romelmac gamoiyena p. papkoviCis (1940) ganzogadoebuliorTogonalurobis Tanafardoba; es Tanafardoba papkoviCis miermiTiTebuli iyo erTgvarovan amonaxsnTa funqciebis sasazRvropirobebisaTvis, romlebic Seesabamebian TviT am funqciebis damaTi pirveli warmoebulebis nulad gadaqcevas fenis paralelurgverdebze; papkoviCis meTodis mkacri dasabuTeba mocemuli iyomogvianebiT g. grinbergis mier (1953).mrgvali filis wonasworoba nebismieri RerZulsimetriulidatvirTvisas gamokvleuli iyo erTgvarovani amonaxsnebisdaxmarebiT v.prokopovis mier (1958). mrgvali filisRerZulsimetriuli Runva zogadad ganxilulia b. abramianisa daa. babloianis mier (1958); zusti amonaxsni amocanisa, gverdiTizedapiriT Camagrebuli filis wonasworobis Sesaxeb, algebrulgantolebaTa usasrulo sistemis saSualebiT mogvces v.grinCenkomda a. ulitkom (1963). analogiuri Sedegebi miiRo g. valovma (1962).sqeli filebis RerZulsimetriuli Runvis zogierTi kerZoSemTxveva ganxilulia n. glazunovas (1963) mier. a. babloianma(1964) gamoikvlia mrgvali filis araRerZulsimetriuli datvirTva,roca gverdiT zedapirze mocemulia gadaadgilebebi (amonaxsniwarmodgenilia ormag mwkrivebSi, romelTa koeficientebi iZebnebausasrulo sistemebidan).usasrulo sqeli fila mrgvali naxvretiT ganixileboda o.aqsentianis (1965) naSromSi; erTgvarovani amonaxsnebis gamoyenebamsaSualeba mogvca gadagvewyvita amocana xvrelis axlos Zabvebiskoncentraciis Sesaxeb, gantolebaTa usasrulo sistemaze dayvaniTkoeficientebisaTvis erTgvarovani amonaxsnebiT; m. abenovam (1965)msgavsi amocana daiyvana fredholmis tipis integralurgantolebebamde.37

arastacionaruli amocana sqeli filis Termodrekadi(kvazistatikuri) wonasworobis Sesaxeb ganxilulia a. Sevelevis(1965) mier. r. rapoportma (1962) miiRo miaxloebiTi erTgvarovaniamonaxsnebi sqeli filisaTvis, romlebic agebulia rodesac argvaqvs ganivi deformaciebi. ukanasknel daSvebas mivyavarTorTogonalur sakuTar funqciebamde.usasrulo cilindris drekadi wonasworoba Seiswavlebodabevri avtoris mier. RerZulsimetriuli amocana Rru cilindrzenormaluri wnevis moqmedebis Sesaxeb, romelic modebuliagverdiTi zedapiris ubanze, ganxiluli iyo 1943 w.-s g. Sapirosmier; es amocana man gadawyvita furie-beselis integralebissaSualebiT (es gadawyveta mogvianebiT gaimeora v. popovma, 1956).mTliani da Rru cilindrebisaTvis erTgvarovani amonaxsnebimaTi RerZulsimetriuli deformaciebisas ganixileboda v.prokopovis mier (1949, 1950) [20]. RerZulsimetriuli amocanausasrulo mTliani cilindrisaTvis, romelic datvirTulianormaluri ZalvebiT gverdiT zedapirebze, Seswavlili iyo a.luries (1953) mier; am amocanis amonaxsni warmodgenili furiesintegralebis formiT konturuli integrebis meSveobiT,gamosaxulia funqciebiT, romlebic Seesabamebian cilindrisSesaxeb amocanis erTgvarovan amonaxsnebs. zRvruli gadasvliTmiRebulia amocanis amonaxsni `Semosartyluli~ cilindrisSesaxeb. mxebi datvirTvis SemTxveva da aseve zedapiruli ZalvebiTusasrulo cilindris Runvis Sesaxeb amocana, gamokvleulia imavemeTodiT p. livSicis (1960, 1963, 1964) statiebSi.usasrulo cilindris gverdiT zedapirebze moqmedi rTulidatvirTvis SemTxvevaSi, roca datvirTva warmoidgineba furiesintegraliT RerZul koordinatze da furies mwkrivebiT kuTxeze,gamokvleuli iyo k. solianik-krasis mier (1960) [21]. mis mierveaganxiluli ufro zogadi amocana brunvis tanis wonasworobisSesaxeb, roca meridianuli kuTxis trigonometriuli funqciebiSeiZleba iyos gamoyofili amonaxsnSi calkeuli TanamamravlebissaxiT (1958); Rru cilindrisaTvis mis mier iqna gamokvleuli38

(1965) datvirTvis gavlena, romelic ganawilebulia gverdiTzedapirze ϕ kuTxis mimarTulebiT nebismieri saxiT da romelicwarmoadgens polinoms RerZuli z koordinatiT (torsebzesruldeboda integraluri pirobebi).Sereuli RerZulsimetriuli amocana usasrulo mTliani anRru cilindrisadmi ganixileboda b. koganis, a. xrustaliovis, f.vainSteinis (1958, 1959, 1963) statiebSi; liavis ZabvaTa funqciaigeboda maT mier konturuli integralis saxiT, romelicSeicavda saTanadod SerCeul funqciebs, damokidebulebserTgvarovani amonaxsnebis parametrebze cilindrisaTvis; b.koganisa da a. xrustaliovis naSromSi (1959) gamoyenebuliawyvil integralur gantolebaTa meTodi.Rru Tu mTliani sasruli cilindris wonasworobaRerZulsimetriul SemTxvevaSi Seiswavleboda erTgvarovaniamonaxsnebis saSualebiT v.prokopovis (1950, 1958) mier; g.buxarinovma (1956) daiyvana sasruli sigrZis mTliani cilindrisRerZulsimetriuli deformaciis Sesaxeb amocanis amoxsnadamatebiTi funqciis moZebnamde, romlisTvisac igeba integrodiferencialurigantoleba. ormocdaaTiani wlebis SemdgomgamoCnda bevri naSromi, miZRvnili sasruli sigrZis mTlianicilindris _ wonasworobis amocanisadmi, romlebSic amocanisamoxsna dadis wrfiv algebrul gantolebaTa usasrulosimravleze (b. abramiani, 1954, g. valovi, 1962; v. lixaCovi, 1965).mrgvali cilindris kumSvas ikvlevdnen g. miroSniCenko (1957) da g.valovi (1961) (1957); mbrunavi cilindris wonasworoba ganixila v.grinCenkom (1964); mis miervea mocemuli detaluri analiziRerZulsimetriul amocanaze sasazRvro pirobebis yvela aspeqtiszusti Sesrulebis Sesaxeb naxevrad usasrulo cilindrisaTvis(1965). sasruli sigrZis cilindris RerZulsimetriulideformacia, romelic gakeTebulia transversalurad izotropulimasalisagan, Seiswavleboda a. babloianis mier (1961).calkeul SemTxvevebSi xerxdeba, rom dakmayofildes yvelasasazRvro piroba sasrulo sigrZis cilindris wonasworobis39

Sesaxeb amocanaSi, ise rom ar mivmarToT am dros usasrulosistemaTa amonaxsnebs (ix. b. abramiani, 1958; g. valovi, 1957, 1958).cilindris zedapirebze yvela sasazRvro pirobebiserTdrouli zusti Sesrulebis sirTulem gvaiZula veZeboTamocanis amoxsnis miaxloebiTi gzebi; ase, s. treninma (1952) [22]warmoadgina daZabuli mdgomareoba ori tenzoriT: ZiriTadiT damakoreqtirebeliT, amasTan ukanaskneli ar iZleva ZabvebsgverdiT zedapirebze (erTgvarovani amonaxsni) da misiparametrebi ganisazRvreba energetikuli gziT. ufro zogadi (araRerZulsimetriuli) amocana Rru cilindrze ganixilebodaanaloguri saxiT v. ionovis (1957) mier; s. Sainma (1962) mogvcamakoreqtirebeli tenzoris ageba pirvel miaxloebaSi.sqelkedliani cilindris araRerZuli deformacia iyoSeswavlili maTi mwkrivebSi warmodgenis saSualebiT, romlebicSeicavdnen beselisa da makdonaldis funqciebs i. smolovikis daa. Sepetevis (1961) naSromSi da v. sumcovis (1957-1959) naSromebSi.Rru cilindris datvirTvis zogad SemTxvevaSi sasazRvropirobebis mkacri Sesruleba, romelTac mivyavarT usasrulosistemebamde iyo ganxorcielebuli e. baidas (1959, 1960) mier.a. kvitkas (1959) statiebi miZRvnilia xerxebis SemuSavebisadmi,romlebSic saSualeba mogvecemoda sqeltaniani cilindrisRerZulsimetriul deformaciaze gamokvlevebi dagveyvanagamomTvleli manqanebis gamoyenebaze.f. goxbaumma (1964) gamoiyena a. luries simboluri meTodimTliani da Rru cilindrebis mimarT, romlebic ganicdidnenZiriTadad RerZuli simetriuli datvirTvis moqmedebas.Rru (da mTliani) cilindrebis gaangariSebis miaxloebiTimeTodi maTi RerZulsimetriuli datvirTvisas iyoSemoTavazebuli v. bidermanis (1946; 1950) mier; warmoadgenda ramxeb Zabvebs RerZuli da radialuri funqciebis namravlis jamissaxiT, bidermani iRebda radiusis Sesaferis funqciebs, xoloRerZuli funqciebisaTvis Rebulobda potencialuri energiisminimumis principidan gamomdinare Cveulebriv diferencialur40

gantolebebs, romlebic Seicavdnen marjvena nawilebSifunqciebs, damokidebuls cilindris gverdiT zedapirebzemodebul normalur Zalebze. meTodis gavrcoba mxebi Zalebisarsebobis SemTxvevaze ganaxorciela SemdgomSi v. gorskim (1963).Rru cilindrebis gaangariSebis sxva miaxloebiTi xerxi,romlebic datvirTulia gverdiT zedapirze, miTiTebulia s.boiariSnikovis (1953) mier, romelmac SemogvTavazagadaadgilebebisaTvis gamosaxulebis gamoyeneba, rac Tavis mxrivwarmoadgens Txeli drekadi garsebis TeoriaSi gamoyenebulimeTodebis ganzogadebas. TandaTanobiTi miaxloebis originalurimeTodi, misadagebuli amocanasTan cilindris wonasworobisSesaxeb, SeimuSava f. detinkom (1953) [23]; man aago amonaxsnimwkrivebSi mcire parametris xarisxebad (puasonis koeficienti).stacionaruli amocana Rru cilindris (RerZuli simetriisSemTxvevaSi) Termodrekaduli wonasworobis Sesaxeb Seiswavlebodajer p. ogibalovis (1954), xolo Semdeg i. SevCenkos (1958) mier,romelmac gaiTvaliswina masalis drekadobis modulis cvlilebacilindris RerZis gaswvriv. a.podgornim (1965) gaiTvaliswinacilindris torsebis gavlena da aseve centridanuli Zalebisgavlena; amocana amoxsnilia miaxloebiT, lagranJis variaciuliprincipis gamoyenebiT. p. ermakovma (1961) da v. SaCnevma (1962)ganixiles Termodrekadobis stacionaruli amocana sasrulisigrZis mTliani cilindrisaTvis misi RerZulsimetriulideformaciisas; am naSromTagan pirvelSi pirobebi torsebzesruldeba miaxloebiT, Tanaxmad bidermanis meTodisa, xolomeoreSi – amocanis amoxsna dayvanilia integralur-diferencialurigantolebis amoxsnamde. Termodrekadobis stacionaruli amocanausasrulo cilindrisaTvis ramdenime RruTi formulirebulia a.kosmodamianskis (1962) [24] mier – rogorc temperaturuli veli,aseve temperaturuli mdgomareoba ganisazRvreba bubnovgaliorkinismeTodiT.Termodrekadobis arastacionaruli amocana mbrunavi RrucilindrisaTvis Seiswavleboda i. SevCenkos (1961) mier, romelic41

asrulebda pirobebs torsebze, kastilianos variaciuli meTodissaSualebiT. a.Sevelevi (1966) xsnida Termodrekad amocanasusasrulo cilindrisaTvis, romlis garSemoc temperaturaicvleba eqsponencialuri kanoniT drois mixedviT. is sazRvravdamaqsimaluri Tburi Zabvebis damokidebulebas gacxelebisparametrebze, rac saSualebas iZleva CamovayaliboT optimaluriamocana. a. uzdaliovma (1962) [25] ganixila Termodrekadobisarastacionaruli brtyeli RerZulsimetruli amocanaanizotropuli mTliani da Rru cilindrebisaTvis.erTgvarovani amonaxsnebi Rru sferosaTvis, misi simetriulideformaciis SemTxvevaSi iyo miTiTebuli 1943 wels a. luriesmier; am amonaxsnebis gamoyenebam saSualeba mogvca amogvexsnahipoTezebis saSualebiT sfero, romelic moWrilia konusurizedapiriT da romelsac aqvs mwvervali centrSi erT an orivepolusTan; luriem Caatara agreTve im amonaxsnebis sizustisSefaseba, romlebic dafuZnebulia kirhxof-liavis kinematikurihipotezebiT sargeblobaze sferul garsTan mimarTebaSi.Rru sferos wonasworobis Sesaxeb amocana misi nebismierideformaciisas amoxsnilia a. luries (1953) mier p. papkoviCiszogadi amonaxsnis saSualebiT; meoTxe funqciis marjve arCevisada harmoniuli veqtorebis wyalobiT avtorma moaxerxa arsebiTadSeekveca gamoTvlaTa moculoba Rru sferosaTvis rogorc meoreZiriTadi amocanis SemTxvevaSi, aseve pirveli ZiriTadi amocanisSemTxvevaSic. drekadobis Teoriis sivrciTi amocanebisgamokvlevaTa Sedegebi Tavmoyrilia mis monografiaSi (1955),sadac Sesulia aseve amocanebis amonaxsnebi mZime da mbrunaviburTis Sesaxeb sferuli Rrus SemousazRvrel garemoSi da sxva.sferos Sesaxeb amocanis amoxsnis meore xerxi, dafuZnebulidrekadobis Teoriis brtyel da RerZulsimetriul amocanebsSoris kavSirze da analizur funqciaTa Teoriis gamoyenebaze,SemogvTavazes a. aleqsandrovma da i. soloviovma (1962) [26],kumSva, sufTa Runva da Runva im Rru sferosi, romelic moWrilia42

polusebis konusuri zedapirebiT ganxilulia k.solianik-krasas(1962) [27] mier;daZabuli mdgomareoba sferul sartyelSi, romelicimyofeba Sida wnevis moqmedebis qveS, Seiswavleboda a. ulitkos(1962) mier.amocana vertikaluri cilindruli siRruvis siaxloves mZimedrekadi masivis daZabuli mdgomareobis Sesaxeb pirveladdasmuli iyo a. dinikis (1925) [28] mier, samTo qanebis wnevisSesaxeb sakiTxTan kavSirSi; ufro dawvrilebiT es amocanaSemdgomSi Seiswavleboda s. lexnickis (1938, 1940) mier, maTricxvSi transversalur izotropuli naxevarsivrcisaTvisac.Zabvebis koncentraciaze cilindruli siRruvis gavlenasivrciTi daZabuli mdgomareobisas gamoikvlia s. gutmanma (1960).cilindruli siRruvis zedapiris ubanze modebuli gare ZalebisgaTvaliswineba moaxdina g.CakvetaZem (1956, 1959); mis sxvanaSromebSi ganixileboda drekadi naxevarsivrce sferuli (1955)da cilindruli (1956) siRruveebiT; gamokvlevis misi xerxidafuZnebulia RerZulsimetriul amocanaSi kompleqsuri cvladisSemotanasa da n. musxeliSvilis meTodebis misadagebaze. sferulisiRruvis siaxloves ZabvaTa koncentracia mZime naxevarsivrceSigamoikvleoda n.fleiSmanisa da v. gnatikovis (1954) mier.r. kaufmanma (1958) [29] ganixila sferuli siRruvisSemcveli drekadi fenis amocana; misi amoxsnis meTodi Sedgebasferuli koordinatTa sistemis saTavis gadatanaSi da sferulfunqciebisaTvis gadatanis formulebis SemotanaSi; sxva statiaSikaufmanma imave meTodiT amoxsna amocana sferos wonasworobisSesaxeb arakoncentrirebuli sferuli RruTi. p.perlinma (1964)aago meore ZiriTadi amocanis amonaxsni Rru brunviTielifsoidis Sesaxeb, romlis Sida zedapirs warmoadgens sfero;i.podilCukma (1965) Seiswavla sferul koordinatebSi Sida dagare amocanebi brunviTi elifsoidisaTvis. aq naxseneb oTxnaSromSi amonaxsni igeba mwkrivebSi, romelTa koeficientebiunda ganisazRvron gantolebaTa usasrulo sistemidan.43

v. Jarovma (1963) dasva mniSvnelovani amocana TermodrekadiZabvebis Sesaxeb magravitirebel sferoSi temperaturis ganawilebisnebismieri kanonisas; Termodrekadobis stacionaruli amocanaRru sferosaTvis, romlis modulic aris radiusis xarisxovanifunqcia, gadawyvita i.danilinam (1962).sakiTxi konusis (mTliani da Rru) wonasworobis SesaxebRerZulsimetriuli datvirTvis moqmedebisas ganixila g. Sapirom(1944); man miiRo amocanis polinomuri amonaxsni zedapirulidatvirTvebis zogierTi tipisa da simZimis Zalis moqmedebisaTvis;sxvagvari xerxiT es amocana gamoikvlia a.aleqsandrovma (1962)[30]. konusis mwvervalze modebuli Seyursuli momentis moqmedebaganixila a. ulitkom (1960); mis sxva naSromSi (1960) melinisgardaqmnebis saSualebiT ixsneba zogadi amocana drekadikonusis wonasworobis Sesaxeb. RerZulsimetriulad datvirTulikonusis drekadi wonasworoba ganixila agreTve k. solianikkrasma(1955, 1962); amonaxsni man warmoadgina furies integralebissaxiT. v. ionovma (1965) mogvca amocanis amonaxsni konusuri tanisRerZulsimetriuli deformaciis Sesaxeb; sasazRvro pirobebisSesrulebas mivyavarT gantolebaTa usasrulo simravlisakenmakoreqtirebeli tenzoris mudmivebisaTvis. zedapiruladdatvirTuli konusis grexa ganxilulia k. solianik-krasas (1965)da p.nedorezovis (1965) mier.mZime brunviTi paraboloidis wonasworobis Sesaxeb amocanagadawyvetilia g. Sapiros (1950) [31] mier; paraboloidis gaWimva daRunva, da aseve tanis, romelic Seicavs paraboloidur Rrus,gaWimva da Runva ganxilulia k. solianik-krasas (1958) mier; missxva naSromSi (1958) gamokvleulia elifsoidisa da calRruianihiperboloidis kumSva; hiperboloidis grexa gamoikvlies n.lebedevma da s. skalskaiam (1966).toroidaluri koordinatebis saSualebiT a. zaxareviCma (1952)[38] ganixila brunviTi toris wonasworoba; v. levSinma (1962) aagogare da Sida wnevis moqmedebis qveS moqceuli Rru toris amocanisamonaxsni. mrgvali ganivi kveTis mqone toris grexa, xraxnuli44

zambarebis gaangariSebasTan kavSirSi, romelTac xveulTa mcirebiji aqvT, dawvrilebiT Seiswavla k.solianik-krasam (1950);amonaxsni man miiRo bipolaruli koordinatebis gamoyenebiT daigi Seicavs mwkrivebs, romlebSic hiperboluri, trigonometriulifunqciebi da mierTebuli leJandris funqciebia CarTuli.mcire elifsoidaluri Rrus Semcveli mrgvali RerosgaWimva gamokvleulia k.solianik-krasis (1958) mier elifsoidalurikoordinatebis gamoyenebiT. n. forsmanma (1958) [32] gadawyvitaamocana mrgvali ganivi kveTis mqone gaWimul ReroSi Zabvebiskoncentraciis Sesaxeb, romlebic sisqis cvlilebis adgilSia:miRebulia gansazRvruli integralebis amonaxsni, romlebicSemdeg gamoiTvleba miaxloebiTi formiT.1963 wlidan gamoCnda i. voroviCisa da misi mowafeebisnaSromebi miZRvnili filebisa da garsebisaTvis asimptoturiamonaxsnebis agebisadmi, amasTan aseTi agebis safuZvlad miiRebadrekadobis Teoriis samganzomilebiani amocanis SesabamisierTgvarovani amonaxsnebi; lagranJis variaciuli meTodiT dgebagantolebaTa usasrulo sistemebi saZiebeli funqciebiskonturuli mniSvnelobebisaTvis; am sistemaTa amonaxsni igebamwkrivebSi filis an garsis sisqis xarisxebis mixedviT. ammeTodiT gamokvleulia filis Runvis amocana (o. aqsentiani da i.voroviCi (1963, 1964), da aseve cilindruli da sferuli garsebisSesaxeb RerZulsimetriuli amocanebi (n. bazarenko da i.voroviCi,1965; t. vilenskaia da i. voroviCi, 1966) [33].lames klasikuri amocana marTkuTxa paralelepipediswonasworobis Sesaxeb, romelic datvirTulia waxnagebzemocemuli ZalvebiT, ipyrobs bevri mkvlevaris yuradRebas,dawyebuli m. filonenko-borodiCis Sromebidan. am mimarTulebispirvel statiaSi, romelic 1946 wels gamoqveynda, m.filomenkoborodiCmaSemoitana ganxilvaSi kosinus-binomi – mimdevrobaaraorTogonaluri funqciebisa, romlebsac gaaCniaT sisrule maTigansazRvris intervalSi da iqcevian nulad Tavis pirvelwarmoebulebTan erTad intervalis sazRvrebze.45

m. filonenko-borodiCis Semdgom naSromebSi kosinus-binomimis mier gamoiyeneboda marTkuTxa paralelepipedis wonasworobisSesaxeb amocanaSi miaxloebiTi amoxsnisaTvis. amocanis amoxsnisidea Sedgeboda ZabvaTa tenzoris or nawilad gayofaSi: ZiriTaditenzori, romelic akmayofilebs wonasworobis gantolebebsa daparalelepipedis waxnagebis datvirTvis pirobebs damakoreqtirebeli tenzori, romelic aigeboda kosinus-binomebisa damaTi warmoebulebis saSualebiT. ukanaskneli tenzori, romelicakmayofilebda ra wonasworobis gantolebebsa da nulovansasazRvro pirobebs, Seicavda nebismier mudmivebs, romlebicganisazRvreba kastilianos variaciuli meTodiT. m. filonenkoborodiCma(1951) Seiswavla amocana paralelepipedis toli dasapirispirod mimarTuli datvirTvebiT SekumSvis Sesaxeb daganixila paralelepipedis Termodrekadi wonasworoba; mogvianebiT(1953) man ganavrco es meTodi cilindruli koordinatebisSemTxvevaze: masve ekuTvnis mosazreba ZiriTadi tenzorisSerCevis Sesaxeb nebismierad datvirTuli paralelepipedisaTvis.drekadi paralelepipedis wonasworobis Sesaxeb amocanisamoxsnisadmi sxva midgomaa ganviTarebuli b. bondarenkos (1961,1963) naSromebSi, romelic iyenebda gadaadgilebebSi drekadobisTeoriis gantolebebis polinomur amonaxsns, amasTan nebismierikoeficientebi am amonaxsnebSi ganisazRvreba umciresi kvadratebismeTodiT.zogierTi kerZo amocanebi marTkuTxa paralelepipedisaTvis,gadawyvetili mwkrivebSi, ganxilulia g. valovis (1959),a.melqonianis (1960), a .babloianis da s. saakianis (1960) mier.paralelepipedis wonasworobis Sesaxeb amocanis Seswavlas,usasrulo sistemebis saSualebiT, eZRvneba e. vaidas (1958, 1959)ori statia. ufro dawvrilebiTi gamokvlevebi paralelepipediswonasworobis Sesaxeb, sxvadasxva tipis datvirTvebisa dasxvadasxva sasazRvro pirobebisaTvis usasrulo sistemebissaSualebiT, Catarebulia g. balovis (1957-1959, 1966) s.saakianis(1965), a. babloianis da s. saakianis (1964) SromebSi; am ciklSi46

ganxilulia rogorc pirveli da meore ZiriTadi amocanebi, iseSereuli da sakontaqto amocanebis zogierTi tipebi, amasTangansakuTrebuli yuradReba eTmoba miRebuli usasrulosistemebis regularobis (anu kvazi-sruli-regularobis)damtkiceba.1.1-is daskvnebiliteraturis mimoxilvidan SeiZleba Semdegi daskvnebisgakeTeba:− samecniero literaturaSi praqtikulad ar gvxvdebaTxelkedliani sivrciTi sistemebis gaangariSebis meTodebisistoriuli analizi;− zogierTi Sromebis gamoklebiT, ganzogadoebuli impulsurifunqciebi gamoiyenebian mxolod diferencialuri gantolebebisCawerisaTvis, magram ara maTi amoxsnebis misaRebad. amave droswyvetili funqciis SemoReba iZleva gaangariSebis iseTiprincipulad axali meTodebis miRebis saSualebas, romlebicafarToeben amoxsnadi amocanebis klass da anzogadeben yvelaamoxsnad amocanebs erTian meTodologiur safuZvelze.drekadobis wrfivi Teoriis istoriis safuZvelze dReisaTviskidev ufro meti istoriuli masalis gadmocema da misianalizia gasakeTebeli.47

2. შედეგები და მათი განსჯა2.1. sen-venanisa da almanzis amocanebiswarmodgenarogorc cnobilia, prizmuli Reros Tavisufali grexisamocana daiyvaneba harmoniul problemaze, romlis amoxsnaTameTodebi kargadaa SemuSavebuli. adreuli naSromebi Reroebisgrexaze miZRvnilia am amocanis Caketili saxiT antrigonometriuli mwkrivebis saSualebiT amoxsnisadmi; maTmiekuTvneba b. galiorkinis statiebi, romlebSic gamokvleuliatolferda marTkuTxa samkuTxedis formis ganivi kveTis mqoneprizmis grexa (1919) [3] da paraboluri ganivi kveTis prizmebi(1924) [3]; amocana rigi kveTebis grexis Sesaxeb, romlebicSemosazRvrulia algebruli mrudebiT, gadawyvetilia d. benovis(1935, 1937) da d. gavras (1939) [34] naSromebSi; mogvianebiTparaboluri prizmebis grexas ikvlevda v. bloxi (1959). mTlianian Rru lilvebis grexisas radialuri bzaris gavlenaSeswavlilia a. lokSinis (1928) da v. liskovis (1930) statiebSi.grexis Teoriis amocanebis amoxsnis sxvadasxva meTodebisadmi,(eqsperimentaluri meTodebis CaTvliT), miZRvnilia a. dinikismonografia, romelic gamoqveynebulia 1938 wels.1925 wels g. kolosovma da d. gavrma grexis amocanisamoxsnisas pirvelad gamoiyenes kompleqsuri cvladebi; maTganixiles amocana mcire centraluri kuTxiT arawriuliseqtoris grexis Sesaxeb. am mimarTulebiT fundamenturiSedegebi miiRo n. musxeliSvilma (1929), romelmac aCvena, romcalmbuli da oradbmuli midamos grexis amocana, romelicasaxavs mocemul midamos Sesabamisad wresa da wriul rgolze,daiyvaneba kompleqsuri cvladis funqciis moZebnaze. prizmuliReroebis grexis amocanebis amoxsnisas kompleqsuri cvladisfunqciis Teoriis meTodebi gamoiyeneboda aseTi Reroebissxvadasxva profilebisaTvis d. avazaSvilis (1940), a. batirevis48

(1953), x. muStaris (1938), a. ugodCikovis (1956) da sxvaTa statiebSi.r. kuzminma (1946) gamoiyena konformuli asaxva sxvaformaSi; man miiRo dasagrexi Reros sixistis uSualogamoTvlisaTvis moxerxebuli formula. am formulam saSualebamogvca gamogveTvala sixiste im profilebisaTvis, romelTakonturi Seicavs kuTxis wertilebs. sxva naSromSi, romelicekuTvnis p. kufariovs (1937), kompleqsuri cvladis funqciaTameTodi vrceldeba iseT konturis SemTxvevaze, romelic SeicavskuTxis wertilebs. kufariovis meTodi o. babakovma (1954)gamoiyena zeturi profilis grexis ganxilvisas.konformuli asaxvis meTodiT e. Siriaevma ganixila lilvisgrexa, romelsac radialuri da aseve grZivi rkaluri bzari aqvs(1956). sxva naSromSi Siriaevma gamoikvlia mrgvali lilvis grexa,romelsac sxvadasxva siRrmis Wrili aqvs (1958). wriuliamonaCarxebis mqone lilvis grexas swavlobda a. skorobogatko(1958, 1962) [35]. Rru saaviacio profilebis grexa, kompleqsuricvladis funqciis Teoriis saSualebiT ganixila g.tirskim (1959).kuTxovanisebri, jvarisebri da tesebri profilebis grexisamocanis miaxloebiTi amonaxsni konformuli asaxvis saSualebiTmiiRo v.maxovikovma (1957). ganaviTara ra konformuli asaxvismiaxloebiTi xerxebi, a. ugodCikovma (1956) [36] ganixila mrgvalililvis grexa, romelsac kbilakebi (Rarobebi) aqvs da milovanililvi, romelsac Siga kbilakebi (Rarobebiani quro) aqvs.Rru Reroebis grexisa da Runvis amocanis amoxsnis axalixerxi 1948 wels SemogvTavaza d. Sermanma [37]. man Semoitanadamxmare funqcia, romelic dakavSirebulia oradbmuli midamoserT-erT sazRvarze grexis kompleqsur funqciasTan raRacTanafardobiT; es damxmare funqcia akmayofilebs fredholmisintegralur gantolebas, romlis amoxsnac daiyvanebakvaziregularul (xolo xandaxan sruliad regularul) wrfivalgebrul gantolebaTa usasruli sistemis amoxsnamde. Sermanmaam xerxiT amoxsna wrewirebiTa da elifsebiT (1950, 1951, 1953)49

SemosazRvruli oradbmuli profilebis grexis rigi konkretuliamocanebi.Semdgomi Teoriuli Ziebani am mimarTulebiT, romlebmacmigviyvanes Rru Reroebis grexis rigi amocanebis amoxsnamde,tardeboda d. Sermanis (1953, 1955, 1969) [38], r. stepanovis da d.Sermanis (1952) [39], i. amenzades (1958) [40] mier. Sermanis meTodi iyogamoyenebuli l. kapanianis (1952) [41], v.iakovlevis (1956) daagreTve i. baxtiarovis (1959) naSromebSi kolofa ReroebisgrexisaTvis, m. ismailovis (1959) naSromSi ki samkuTxa prizmuliRrus mqone mrgvali lilvis grexisaTvis. m. neimanis (1958)naSromSi mravalwaxnaga TanaRerZuli Rrus mqone mrgvalililvis grexis amocanaSi.prizmuli Reroebis, romelTac aqvT gadamkveTi wrewirebisori rkaliT SemosazRvruli ganivi kveTi, grexisa da Runvisamocanebis zusti amonaxsnebi, miRebuli iyo 1949 wels i.ufliandis mier bipolaruli koordinatebis gamoyenebiT; immidamoebisaTvis, romlebic uSveben amonaxsns bipolarulkoordinatebSi, Runvisa da grexis amocanebis amoxsnisdawvrilebiT gadmocema moyvanilia mis monografiaSi (1950) [42].mogvianebiT v. bloxma (1956) gamoaqveyna statia, miZRvniliorTogonaluri wrexazebis rkalebiT Seqmnili marTkuTxedisgrexis amocanisadmi bipolarul koordinatebSi.linzisebri kveTis mqone Reros grexas ganixilavdneni.buraki da m.leonovi (1960) [43]. s. gridnevma bipolarulikoordinatebi gamoiyena oradbmuli profilis grexis amocanisSeswavlisas (1963) da daiyvana aseTi amocanis amoxsna usasrulosistemebamde.rgolis seqtoris midamosaTvis grexis amocanisamoxsna miiRo k.kitoverma (1954) [44] elifsebisa da hiperbolebisrkalebiT Seqmnili rigi midamoebisaTvis. grexis amocanis zustiamonaxsnebi elifsur koordinatebSi miiRo v. bloxma (1964).Reroebis grexisa da Runvis Sesaxeb amocanis amoxsnismiaxloebiTi meTodebi SemuSavebulia d. panovis mier (1934, 1936,50

1938) [45]. is anviTarebda mcire parametris da grafikul meTodebs,Seiswavlida prizmatulTan axlo myofi Reroebis grexas,xraxnuli profilis grexasa da Runvas; man agreTve ganixilaortesebri koWisa da sogmanis mqone lilvis grexis amocanasasrul sxvaobaTa meTodiT.grexis TeoriaSi m. slobodianskis naSromebSi (1939, 1940, 1951)[46, 47, 48] sasrul sxvaobaTa meTodi gamoyenebuli iyo mxoloderTi cvladiT da amocanis amonaxsni dadioda Cveulebrivdiferencialur gantolebaTa sistemaze; am meTodma saSualebamisca slobodianskis, xolo Semdeg a.pivovarovs (1953) [49]gamoeTvalaT poligonaluri profilebis Semaval kuTxeebSikoncentraciis koeficientebi. analogiuri meTodi iyo gamoyenebuliv. fadeevas (1949) mier trapeciodaluri kveTis mqone Rerosgrexis Sesaxeb amocanis amoxsnisas. naglini kuTxovanas grexisSesaxeb amocana Seiswavla b. lopovokma (1952); b. roxovskaiam(1946) sasrul sxvaobaTa meTodiT ganixila naglini profilebis(kuTxovana, Sveleri da ortesebri) grexis amocana; mis sxvanaSromSi (1956) da agreTve e.obolenskis (1959) statiaSi es meTodigamoyenebulia Rarobebis mqone lilvis grexis amocanisamoxsnisaTvis.prizmatuli Reroebis Runvisa da grexis Sesaxeb amocanebissxva miaxloebiT meTodebs Soris umniSvnelovanesi adgiliukaviaT variaciul meTodebs, romlebmac didi popularobamoipoves l. leibenzonis da l.kantoroviCis naSromebiswyalobiT. grexis TeoriaSi l. leibenzonis pirvel naSromSi,romelic 1924 wels gamoCnda, Seswavlilia xraxnuli profilisgrexis amocana. masSi miRebulia grexaze xraxnis profilissixistisaTvis miaxloebiTi gamosaxuleba. am formulis Semdgomidazusteba moaxdina v. vetCinkinma (1926) [50].didi mniSvneloba qonda l. leibenzonis (1935) naSromsprizmuli Reroebis Runvis TeoriaSi, romelSic dawvrilebiTaadamuSavebuli am amocanis amoxsnis efeqturi variaciuli meTodida agreTve pirvelad iqna miRebuli Teorema Runvisas mxebi51

Zabvebis cirkulaciis Sesaxeb. Runvis centris moZebnis amocanamSemdgomi ganviTareba hpova n. zvolinskis (1936), d. panovis (1934)da g. proktoris (1936) naSromebSi.l. leibenzonis mravalwliani gamokvlevebis Sedegebi,Reroebis Runvisa da grexis TeoriaSi da agreTve amocanebisamoxsnis efeqturi xerxebis SemuSavebaSi, Sejamebulia mismonografiaSi (1951) [51].[52] 1933 wels l. kantoroviCma SemogvTavaza ormagiintegralis minimumis moZebnis amocanis amoxsnis axalimiaxloebiTi meTodi, romlis Tanaxmad problema daiyvanebaCveulebriv diferencialur gantolebebamde (meTodis krebadobamis mier gamokvleulia mogvianebiT, 1941). p. frumkinTan (1937)erTobliv sxva statiaSi, kantoroviCma warmatebiT gamoiyenaTavisi meTodi rogorc simetriuli, ise arasimetriuli kuTxovanida marTkuTxedis kveTebis grexis amocanebis amoxsnisas. t.Cepovam (1937) [53] ganixila tolgverda trapeciis da agreTvemarTi da iribi simetriuli kuTxovanebis grexa; v. bidermanma(1950) Seiswavla trapeciisa da tolferda samkuTxedis grexa. a.karpovma (1955) mogvca rombis grexis amocanis amonaxsni.a. luriem (1939) [54] miuyena kantoroviCis meTodi simetriulprofils, romelic SemosazRvrulia paraleluri sworebiTa daalgebruli mrudebiT gamosaxuli orwevra gantolebebiT, Runvisada grexis amocanebs. marTkuTxa da tolferda samkuTxedebisgrexis Sesaxeb amocana erTob dawvrilebiT ganixila n.gulkanianma (1953) [55]. specialuri saxis araorTogonalurikoordinatebis SemotaniT n. aruTunianma moaxerxa amoexsnakuTxovanisa da Sveleris grexis amocana (1942); sxva naSromSi manmiiRo izotropuli, an kerZo saxis anizotropiis mqone grexisamocanis amoxsna elifsuri rgoluri seqtorisaTvis.prizmuli Reros grexis amocanis amoxsnis sxva miaxloebiTmeTodze, dafuZnebuls wertilovan interpolaciaze, miuTiTa l.galinma (1939) [56]. Svarcis maalternirebeli meTodiT tesebri52

kveTis mqone Reros grexis amocanis miaxloebuli amonaxsni miiRob. bondarenkom (1956) [57].m. leonovma SemogvTavaza Txelkedliani profilebissixistis gansazRvris miaxloebiTi meTodi, romelicdafuZnebulia toli mxebi Zabvebis `Sua xazebis~ Semotanaze (1956,1957). anviTarebdnen ra am meTods m. leonovma (1957, 1960), g. kitma(1958, 1960) da sxvebma miiRes, rogorc caladbmuli, iseoradbmuli midamoebisaTvis miaxloebiTi amonaxsnebi.h. halimhanovma (1955, 1956) [58] mogvca grexis amocanismiaxloebiTi amonaxsni naTali lilvebisaTvis, romelTa kveTiskonturi Sedgeba ZiriTadi wrexazis rkalebisa da qordebisagan.mis amonaxsnSi mudmivebi ganisazRvreba Zabvis funqciisaganintegralebis nulad qcevis pirobidan, romlebic aRebuliakonturis sworxazovani da rkaluri ubnebis mixedviT. grexisamocanebis SeswavlisaTvis miaxloebiTi meTodebi iyogamoyenebuli agreTve g. sarqisovis da a. amenzades (1952) [59] mier– wesieri mravalwaxnaga profilebisaTvis, l. mitelmanis (1955,1959) mier kvadratisaTvis, naxevarwrisaTvis, tolferdamarTkuTxa samkuTxedisaTvis da saaviacio profilebisaTvis, l.mixailovis (1962) mier naxevarwriuli kveTis mqone RerosaTvis,romelic dasustebulia mrgvali cilindruli RruTi.prizmuli Reros SezRuduli grexis amocana romelsacnebismieri ganivi kveTi aqvs, ganixileboda v. prokopovis (1959)mier, xolo simetriuli profilisaTvis – g. geonjianis (1959) [60]mier; orive naSromSi igulisxmeboda, rom normaluri ZabvebiSezRudul kveTSi Tavisufali grexis deplanaciisproporciulia da gamoiyeneboda amocanis amoxsnis variaciulimeTodi; magaliTisaTvis Seiswavleboda elifsuri da marTkuTxaganivi kveTebi. marTkuTxa ganivi kveTis mqone Reros SezRuduligrexa gamoikvlia agreTve v. netrebkom (1956). man gamoiyena m.filonenko-borodiCis meTodi kastilianos principTan SeTavsebaSi.netrebkos sxva naSromSi (1954) imave xerxiT SeswavliliamarTkuTxa prizmis grexis amocana mis torsebze mxebi Zabvebis53

mocemuli kanoniT ganawilebisas. Rru elifsuri cilindrisSezRudul grexas ganixilavda s. gridnevi (1963).v. galiorkinma 1927 wels mogvca rgoluri seqtoris saxiskveTis mqone prizmuli Reros ZaliT Runvis amocanis zustiamoxsna. Zabvis funqciisaTvis gamosaxuleba man miiRo mwkrivissaxiT. imave naSromSi galiorkinma Seiswavla, mrudwirulikoordinatebis saSualebiT, konsoluri Reros ZaliT simetriuliRunva, romlis profilic SemosazRvrulia parabolebisrkalebiT, parabolebiT da wrfiT, elifsebisa da hiperbolebisrkalebiT. ukanaskneli SemTxveva gamokvleulia agreTve v.tonoianis (1961) statiaSi.d. avazaSvilma (1940) [61] aago konsoluri prizmuli Rerosgrexis amocanis amonaxsni kompleqsuri cvladis funqciissaSualebiT. rgolis midamoze konformuli asaxviT b.obodovskim miiRo elifsuri kveTis Rru Zelis Seyursuli ZaliTRunvis amocanis amonaxsni (1960). l.kapanianma (1956) Runvisamocanis amoxsnisas gamoiyena miaxloebiTi konformuli anasaxi,mrudwiruli kvadratuli amonaWeris mqone wrisaTvis. v.rakivnenkom (1962), ganixila mrgvali cilindris Runvis sakiTxi,rodesac mas aqvs kvadratuli saxis ganivi kveTis mqone orisiRruve.simetriuli Runva Rerosi, romelsac ganivi kveTi marTkuTxamidamoebisagan aqvs Sedgenili, ganixila a. boJenkom (1948). sxvastatiaSi (1954) man Seiswavla naglini profilebis (Sveleri,ortesebri, tesebri) arasimetriuli Runva da gansazRvra Runviscentris mdebareoba. b.gulkanianma (1955) gansazRvra tolgverdatrapeciisa da tolferda samkuTxedis Runvis centriskoordinatebi miaxloebiTi meTodiT. n. popovma mogvca marTkuTxasamkuTxedis saxiT kveTiT prizmis Runvis amocanis amonaxsniCaketili saxiT.d. Sermanma damxmare funqciebis Tavisi meTodi ganavrcoRru prizmuli Reroebis Runvis amocanebze da kerZod, ganixilaelifsuri Zelis SemTxveva, romelic dasustebulia wriuli54

cilindruli RruTi (1953) [62]. Sermanis meTodiT amocanebi RruReroebis Runvis Sesaxeb gamoikvlia i. amenzadem. wre elifsuri(1955) da mrudwiruli (1956) naxvretebiT wre araTanaRerZulielifsuri naxvretiT (1958) [63] da sxva. kveTi elifsis saxiT, oriwriuli naxvretiT, Seiswavla a. kosmodamianskim (1960).[64] 1948 wels n. aruTunianma SemogvTavaza poligonialuriganivi kveTis Reros grexis Sesaxeb amocanis amoxsnis axalimeTodi, romlis arsic mdgomareobs ZabvaTa funqciis moZebnisasdamxmare funqciis SemotanaSi da Semdgom amocanis amoxsnisdayvana sruliad regularul wrfiv algebrul gantolebaTausasrulo sistemamde; SemdgomSi man ganixila kuTxovanis grexisSesaxeb amocana (1949). aruTunianis meTodiT iyo gamokvleulisxvadasxva ganivi kveTis mqone Reroebis grexis Sesaxeb amocanebi;kveTi trapeciis saxiT ganixiles v. abramianma da n. aruTunianma(1951), Sveleri da tesebri e. aleqsandrianma da n. gulkanianma(1953), jvarisebri kveTa da cilindri solisebri kiloTi –b.abramianma (1949, 1959), kolofa profili bzariT – a. babloianma(1958). a. aleqsandrianma (1952) Seiswavla ortesebris, kvadratis damarTkuTxedis, CamoWrili kuTxiT da paralelogramis 45°-ianikuTxiT, SemTxveva, samkuTxa ganivkveTi da marTkuTxedi bzarebiTgamoikvlia n. gulkanianma (1952, 1953); seqtori kbilakebiTganixiles b.abramianma da v. tonoianma (1959).2.1.1. grexisa da Runvis amocanebis dasmada gadawyvetamarTkuTxedis saxis Rru kveTis mqone prizmuli Reroebisgrexa (da Runva) Seiswavla b. abramianma (1950); SemdgomSi mangamoikvlia grZivi siRrueebis mqone mrgvali lilvis SemTxveva(1959); b. abramianisa da a.babloianis naSromSi (1960)gamokvleulia mrgvali Reros grexa amonaCarxebiTa dakbilakebiT, romelsac aqvs centraluri mrgvali siRrue.damxmare funqciebisa da usasrulo sistemebamde dayvanis imave55

meTodiT n.gulkanianma (1960) Seiswavla marTkuTxa prizmis grexa,romelsac ori simetriuli marTkuTxa siRrue aqvs. v. tonoianma(1961) mogvca grZivi amonaCarxebis mqone Rru elifsuri Rerosgrexis amocanis amonaxsni. damxmare funqciis dawvrilebiTimeTodis safuZvliani gadmocema, prizmuli mTliani da RruReroebis, agreTve Sedgenili Reroebisa da brunviTi sxeulebisgrexisas SeiZleba vnaxoT n. aruTunianis da b. abramianis (1963)monografiaSi [65].poligonaluri profilis Reroebis Runvis amocanisadmidamxmare funqciis meTodis gamoyeneba da problemis usasrulosistemebamde dayvana mocemulia n. aruTunianisa da n. gulkanianisstatiaSi (1954). am statiaSi napovnia Runvis centris koordinatebiszusti mniSvnelobebi tesebrisaTvis, SvelerisaTvis daTanabarTaroebiani kuTxovanisaTvis. n.gulkanianma (1959) [66] ipovaaseve Runvis centris koordinatebi kveTisaTvis marTkuTxedissaxiT, romelsac aqvs arasimetriuli marTkuTxa amonaWeri.m. sarqisianma (1956) aruTinianis meTodiT ganixila ortesebrisRunvis amocana. e. kirinma (1963) – gamoikvlia jvarisebri ganivikveTi; v.tonoianma (1961) – elifsis saxis kveTi amonaCarxebiT. a.babloianis (1960, 1961) naSromebi miZRvnilia mrgvali lilvisRunvis amocanisadmi gverdiTi amonaCarxebiT, seqtoruliprizmisa kbilakiT da lilvisa kbilakebiT.n. musxeliSvilma (1932) SeimuSava sxvadasxva masalebisaganSedgenili da gverdiTi zedapirebis gaswvriv erTmaneTTanSekavSirebuli Reroebis grexisa da Runvis Teoria; aseTiamocanis amonaxsni erTmaneTTan SekavSirebuli ori Zelis grexisSemTxvevisaTvis moyvanilia mis cnobil monografiaSi (gam. 2 –1935). i. vekuam da a. ruxaZem (1933) [67] Seiswavles mrgvali ReroTidaarmaturebuli wriuli kveTis cilindris grexa da agreTveSedgenili Reros grexa da Runva, romlis kveTs aqvs fokusurielifsebis saxe; a. ruxaZem (1935) [68] ganixila Sedgenili profilisRunva da grexa, romelic epiTroqoidebiT aris Seqmnili;hipoTroqoidebiT gamijvnis SemTxveva gamoikvlia g. quTaTelaZem56

(1956) [69]. ori wriuli segmentis saxis Sedgenili Reros grexa,romlebic SekavSirebulia qordis gaswvriv, ganixila v. Ziubam daa. asaturianma (1965).Sedgenili Reros grexis Sesaxeb zogad amocanas ekuTvnisk.Cobanianis (1955) statia; masSi moyvanilia Teorema mxebi Zabvebiscirkulaciis Sesaxeb da ganxilulia tesebri ganivi kveTis mqoneSedgenili Reros amocana. k. Cobanianis sxva naSromebSiganxilulia Sedgenili kveTis Runva (1956), Runvis centriskoordinatebis gansazRvra da cvladi diametris mqone Sedgenilililvis grexa (1958) [70]. mravalbmuli Sedgenili Zelis grexagamoikvlia i. suxarevskim (1954) [71]. a. ugodCikovma (1964) ganixilaerTmaneTSi Cadgmuli Sedgenili Reroebis grexa da Runva;amocanis amoxsna igeba konformuli asaxvis saSualebiT dadayvanilia wrfivi gantolebebis usasrulo sistemebze.prizmuli anizotropuli Reroebis grexisa da Runvisamocanebi formulirebuli iqna s. lexnickis naSromebSi (1938,1942, 1956); am gamokvlevaTa Sedegebi da anizotropul garemoTadrekadobis Teoriis rigi sxva amocanebis amonaxsnebiSejamebulia mis monografiaSi (1950) [72]. ufro adre anizotropulprizmaTa grexa ganzogadebuli membranuli analogiis saSualebiTSeiswavla a. lokSinma (1927), ganixilavda ra kveTebs wris,elifsis, marTkuTxedis da paralelogramis saxiT. anizotropuliprizmebis Runvisa da grexis zogierTi amocanebi gamoikvlia l.leibenzonma (1940) variaciuli meTodiT. saaviacio profilisanizotropuli Reros grexis amocanis miaxloebiT amoxsnaseZRvneba v. vantorinis (1939) statia; anizotropiuli Rerosgrexis zogierTi amocanebi miaxloebiTi meTodiT ganixila n.aruTunianma (1947, 1948) [73]. anizotropuli cilindris grexaSeiswavles b. abramianma da a. babloianma (1958).paralelogramis formis ganivi kveTis mqone anizotropuliReros Runva da grexa gamoikvlia r. minasianma (1958) [74].anizotropuli Reroebis Runvis amocanaTa rigi ganixila v.sarqisianma (1961, 1962), romelmac gamoiyena mcire parametris57

xarisxebad mwkrivad gaSlis meTodi. xsnida ra anizotropuliReros Runvis amocanas konformuli anasaxis saSualebiT, e.antonovma (1964) gamosaxa Runvis centris koordinatebi amsaxvelifunqciis koeficientebiT.araerTgvarovani prizmuli Reros grexis amocana amoxsnesb.abramianma (1951) da a. manukianma (1952). v. sarqisianma da v.miqaelianma (1965) Seadgines Sedgenili anizotropuli Reros Runviscentris koordinatebisaTvis formulebi. ukanasknel xanebSigamoCnda Runvis (p.galfaiani, 1960, 1961) da grexis (a. babloiani,1959, p. galfaiani da k.Cobaniani, 1959) [75] amocanebis amonaxsnebisxeulebisaTvis, romlebsac gaaCniaT gamaZlierebeli Txelisafarebi. s. lexnickim Seiswavla grexis zogierTi amocanacvladi drekadobis modulis mqone sxeulebisaTvis (1964, 1965).1950 wels m. bermanma mogvca Runvis centriskoordinatebisaTvis formulebi gamosaxuli funqciebiT,romlebic xsnida imave ganivi kveTis mqone RerosaTvis grexisamocanas; mogvianebiT analogiur Sedegebamde mivida v.novoJilovi (1957) [76]; v. prokopovma (1960) mogvca am formulebisganzogadeba Runvadi Reros mravalbmuli ganivi kveTis SemTxvevaSi.aRniSnuli sakiTxis Semdgomi Seswavla ekuTvnis g. janeliZes(1963). anizotropuli Reros SemTxvevaSi analogiuri SedegebimiiRo v. sarqisianma (1961, 1966). k. Cobanianma da v. miqaelianma(1963) gamoiyvanes formulebi sxvadasxva masalisagan SedgeniliReros Runvis centris koordinatebisaTvis.brunvis sxeulebis grexa Seiswavleboda sxvadasxvagvarimeTodebiT. a.lokSinma (1923) mrudwiruli koordinatebissaSualebiT ganixila konusis, elifsoidis, hiperboloidis dabrunvis paraboloidis grexa; ufro farTo dasmiT brunvissxeulebis grexis Sesaxeb amocana mrudwirul koordinatebSigamoikvlia b. sokolovma (1944); mis miervea ganxilulisafexuriani lilvis grexis amocana ritcis meTodis gamoyenebiT(1939). Rru wakveTili konusis grexa Seiswavla n. paparinma (1937).k. solianik-krasma gamoiyena mrudwiruli koordinatebi im58

lilvebis grexis amocanis amoxsnisas, romlebsac gaaCniaTsiRrueebi (1947) an rgoluri amonaCarxebi (1948, 1955). amgamokvlevaTa Sedegebi Sesulia agreTve mis monografiaSi`cvladi kveTis lilvebis grexa~ (1949) [77]. imave meTodiT manganixila cvladi kveTis mqone RerosYRunvis rigi amocanebi.kerZod, gamokvleulia cilindrul ReroSi sferul siRruesTanZabvebis koncentracia (1955).rgoluri amonaCarxis mqone mrgvali lilvis grexisasZabvebis koncentraciis Sefaseba, romelic kompleqsuri cvladifunqciis Teoriis variaciul meTodTan Sexamebis gamoyenebazeadafuZnebuli, miiRo g. poloJim (1957). badeTa meTodiT, grexisaslilvis diametris mkveTri cvlilebis adgilebSi ZabvaTakoncentraciis Sesaxeb amocana, Seiswavleboda b.rozovskaias (1956,1958) mier. cvladi kveTis mqone milis grexa ganixiles i.amenzademda g. sarqisovma (1959).brunviTi anizotropiuli sxeulebis grexa gamokvleulias.lexnickis (1940), d. grilickis (1957), b. abramianisa da a. babloianis(1958) naSromebSi.mrgvali lilvis gverdiTi zedapiris gaswvriv ganawilebuliZalvebis moqmedeba, romelsac mivyavarT mis dagrexamde,ganixiles n. zvolinskim [79] da p. rizma (1939), romlebmacSeiswavles datvirTvis Tanabari da wrfivi ganawileba. prizmuliReros ufro zogadi SemTxveva ganixiles l. gilmanma da s.goluSkeviCma (1943), p. rizma (1940). l. gilmanis statiaSi (1937)amoxsnilia amocana drekadi rgolis grexis Sesaxeb wyvilebiT,romlebic Tanabradaa ganawilebuli misi RerZis gaswvriv.cilindris msaxvelis gaswvriv Tanabradganawilebuli mgreximxebi Zalvebis SemTxveva Seiswavleboda s.bakanovis (1959) mier.mTliani da Rru wriuli cilindrebis grexa, romlebzecRerZulsimetriulad ganawilebuli zedapiruli datvirTvebimoqmedebs, ganixila furie-beselis mwkrivebis saSualebiT v.bloxma (1954, 1956); imave problemas mTliani cilindrisaTvisubrundeboda p. livSici (1962). s. lexnickim (1961) gadawyvita59

amocana anizotropuli Reros grexis Sesaxeb, romlis gverdiTizedapiris gaswvriv ganawilebulia Zalvebi.safexuriani lilvis grexa datvirTvebiT, romlebicmodebulia gverdiT da torsul zedapirebze da aqvT RerZulisimetria, Seswavlilia b. abramianis da m. jrbaSianis (1951) mier;maT mier amocanis amoxsna dayvanilia wrfivi gantolebebisusasrulo sistemamde. imave meTodiT b.kostandianma gadawyvitaamocana safexuriani Rru lilvis grexis Sesaxeb (1956) [80]; mismiervea ganxiluli marTkuTxedis formis rgoluri amonaCarxismqone lilvis grexa (1954) da lilvis grexa masze dasmulidiskiT (1958). konusuri Reros da cilindruli Reros konusurinawiliT grexa Seiswavla b. abramianma (1958, 1960); n. gulkanianTanTanaavtorobiT mis mier (1961) ganxilulia Sedgenili Rrunaxevarsferos grexa.prizmuli drekadi Reros wonasworobis amocanas, maszeZalvebis moqmedebisas, romlebic modebulia torsebze daTavisufalia datvirTvebisagan gverdiT zedapirebze, uwodebensen-venanis amocanas; drekadobis wrfiv TeoriaSi es amocanaiSleba, bunebrivad, or martiv amocanad (gaWimva da sufTa RunvawyvilebiT), romlebic ixsneba elementaruli gziT da or ufrorTul amocanad (grexa da Runva Zalisagan), romelic ganxiluliazemoT. drekadobis arawrfiv TeoriaSi arsebiTi xdeba urTierTgavlena, gamowveuli sxvadasxvagvari datvirTvebiT; aucilebeliameoradi efeqtebis gaTvaliswineba, romelTa Seswavla dawyebuliiqna 1938, 1939 wlebSi n. zvolinskisa da p. rizis erToblivnaSromebSi. am naSromebis ciklidan ukanasknelSi (1939)ganixileboda gaWimuli Reros grexa. n. zvolinskim SeiswavlaReros grexa, romelic gaWimulia masaTa ZalebiT (1939). gaWimuliReros Runvis amocanas eZRvneba p. rizisa (1939) da a.ruxaZis (1941)gamokvlevebi, romelmac ganixila gavlena Reros Runvaze Runviswyvilis ganivi Zalisagan (1947). meoradi efeqtebi, romlebsacadgili aqvT Sedgenili Reroebis gaWimvasa da Runvisas, iyogamovlenili a.gorgiZisa da a. ruxaZis mier (1943). am60

gamokvlevaTa dazusteba da ganviTareba Catarda Semdgom a.gorgiZis (1955, 1956) [81], r. minasianis (1957), a. ruxaZis (1954), v.mecugovis (1954, 1956) [82] naSromebSi. naSromTa am ciklSi farTodgamoiyeneboda mcire parametrebis meTodi.odnav konusuri da bunebrivad dagrexili Reroebisdeformaciis Sesaxeb amocanebma daikaves mniSvnelovani adgilimecnierTa gamokvlevebSi. aq aseve Zalzed sasargeblo aRmoCndamcire parametris meTodi. pirvelad es meTodi gamoyenebuli iyod. panovis mier odnav konusuri Reros grexis amocanisgadawyvetisas (1938). bunebrivad dagrexili Reroebis gaWimvis,grexis da wyvilebiT Runvis Sesaxeb amocanebi Seswavlili iqna p.rizis (1939) mier. ufro zogad dasmaSi, araorTogonalurikoordinatTa specialuri sistemis gamoyenebiT, sen-venanis amocanabunebrivad dagrexili RerosaTvis, amoxsnili iqna a.luries da g.janeliZis mier (1940) [83]; mogvianebiT g.janeliZem es meTodiganavrco odnav konusur Reroze (1947). dagrexili Reros RunvawyvilZalebiT dekartes koordinatebSi gamoikvlies a. gorgiZemda a. ruxaZem (1944), xolo Runva ganivi ZaliT – a. ruxaZem (1947).Sedgomi gamokvlevebi azusteben da avseben am ZiriTad Sedegebs.dawvrilebiT Seiswavleba meoradi efeqtebi, rTuldebadatvirTvebis sqemebi (a. gorgiZe, v. mecugovi, 1957; a. ruxaZe, 1956;a.Sarangia, 1955) da Sedgenili sustad konusuri Reroebi (s.berZeniSvili, 1957).mcire parametris meTodi warmatebiT iqna gamoyenebulisustad gaRunuli RerZis mqone Reros wonasworobis amocanisamoxsnisas; pirvelad am tipis amocanebi amoixsna p. rizisa (1940,1947) da a. ruxaZis (1942) mier. SemdgomSi ganixileboda gaWimva (r.minasiani, 1954), Runva wyvilZalebiT (a. ruxaZe, 1953) da RunvaZalisagan (a. gorgiZe, 1956).anizotropuli Reros gaWimva da Runva Seswavlili iqna1949 wels s. lexnickis mier. mogvianebiT g. xatiaSvilma ganixilaufro rTuli amocana – anizotropuli Rero sustad gaRunuliRerZiT (1965) [84], mis mierve gamokvleulia sen-venanis amocana61

prizmatulTan axlos Sedgenili anizotropuli sxeulebisaTvis(1963) [85].Reros drekadi wonasworobis amocanas, romlis gverdiTizedapiri datvirTulia ZalvebiT, romlebic warmoadgenenRerZuli koordinatis polinomur funqciebs, ewodeba almanzisamocana. am amocanis kerZo SemTxveva, roca gverdiTi datvirTvaar aris damokidebuli RerZuli koordinatisagan,ASeiswavleboda jer kidev j. miCelis mier. 1960 wels g.janeliZemgamoaqveyna almanzis amocanis amoxsnis zogadi meTodi ZabvebSi,romelsac davyavarT rigi organzomilebiani amocanebis amoxsnamde,romlebic erTmaneTTan dakavSirebulia rekurentuliTanafardobebiT. es meTodi iZleva miCel-almanzis amocanisamoxsnis zogad xerxs da xsnis gzas kompleqsuri cvladisTeoriis gamoyenebisaTvis. miCelis amocanis kerZo SemTxveva,roca gverdiT zedapirze moqmedebs Tanabradganawilebulinormaluri datvirTva Seiswavla a. harisma (1960); man aCvenaRunvis centrebis xazis arseboba, romlis moZebnisaTvissakmarisia sen-venanis amocanis grexis harmoniuli funqciisgansazRvra. Sedgenili ZelebisaTvis miCelisa da almanzisamocanebis amoxsna mogvca g. xatiaSvilma (1953, 1955) [86]. miCelisamocanasTan dakavSirebiT wamoWrili sasazRvro amocanebisklasifikacia da maTi amoxsnis Tanmimdevroba miTiTebulia a.luries mier (1966).transversalurad izotropul cilindrze gverdiTipolinomuri datvirTvis moqmedeba, romelsac mivyavarT misgrexamde da RerZulsimetriul deformaciamde, Seiswavleboda s.lexnickis (1961) mier. a.kosmodamianskim (1956, 1961) ganixilamiCelisa da almanzis amocanebi anizotropuli koWisaTvis. g.janeliZem ganavrco mis mier SemoTavazebuli almanzis amocanisamoxsnis meTodi anizotropuli Reros SemTxvevaze. ufrodawvrilebiT es amocana ganixileboda g. xatiaSvilis mier [81],romelmac gamoikvlia miCelis amocana Sedgenili orTotropulida anizotropuli ReroebisaTvis (1962) da agreTve mogvca62

janeliZis xerxis ganzogadeba almanzis amocanis SemTxvevaSi,Sedgenili orTotropuli RerosaTvis (1964).s.p. timoSenkos 1910 wels [87] gamocemul wignSi „axalimeTodebis gamoyeneba zogierTi xidis konstruqciis mdgradobiskvlevebisas“, ganxiluli aqvs xidebis fermebis elementebisSemowmeba mdgradobaze. kerZod, xazgasmulia, rom fermisgasaTvleli elementis simtkicis uzrunvelsayofad sakmarisi araris misTvis iseTi dasaSvebi daZabulobis SerCeva, rom TavidanaviciloT narCeni deformaciebi an daRlilobis movlenebi.saWiroa agreTve wonasworobis im formis SedarebiTi mdgradobisdamatebiTi gamokvlevebi, romelic safuZvlad udevsgansaxilveli elementis gaTvlebs. mxolod mdgradobis sakmarisimaragis SemTxvevaSi iqneba es elementi mTeli nagebobis mtkiceSemadgeneli nawili.im droisTvis calkeuli elementebis Semowmeba mdgradobazedaiyvaneboda grZivi Runvis formulebis gamoyenebaze.sinamdvileSi gvxvdeboda mdgradobis ufro rTuli sakiTxebismTeli wyeba. naSromSi aRniSnulia, rom grZivi RunvisSemTxvevaSi unda gadaiwyvitos Tamasis saimedoobis sakiTxi.xSirad gvxvdeba mravalmaliani maryuJebis gaswvrivi RunvisSemTxvevebi, xandaxan saWiro xdeba furclebis mdgradobisgansazRvra. calkeuli samagrebi Cveulebriv, yovelgvarigaTvlebis gareSe arian ganlagebuli. CamoTvlilTagan zogierTamocanas gaaCnia Teoriuli gadawyveta da timoSenkom drouladmiiCnia teqnikuri gaTvlebisas, am gadawyvetilebebis praqtikaSidanergva. swored am gaTvlebis saimedoobis donis gazrdiTCndeba imedi, rom momavalSi SesaZlebeli iqneba dasaSvebidaZabulobis ufro maRali normebis miRweva.2.1-is daskvnebi− sen-venanisa da almanzis amocanebSi prizmuli RerosTavisufali grexis amocana daiyvaneba harmoniul problemaze,63

omlis amocanaTa amoxsnis meTodebi kargadaa damuSavebulida gaanalizebuli;− drekadi prizmuli Zelebis daZabul-deformirebuli mdgomareobisdadgena, rodesac Zelis boloebze moqmedebs nebismieriZalTa sistema, aseve warmoadgens drekadobis Teoriis erTerTZiriTad da rTul maTematikur amocanas. maTematikuriTvalsazrisiT igi ar aris bolomde miyvanili da gadawyvetili,Tumca e.w. `sen-venanis principis~ daxmarebiT xerxdebaam amocanis gadawyveta, romelic miaxloebiTia da ar SeiZlebaCaiTvalos zustad. swored, amgvar klasikur midgomadaacnobili sen-venanis mosazreba, romelic literaturaSidamkvidrda `sen-venanis principis~, kerZod ki sen-venanisnaxevrad Sebrunebuli meTodis saSualebiT.2.2. drekadi tanis statikis Sereuli sivrciTiamocanebis Seswavlis istoriadrekadobis maTematikuri Teoriis Sereul amocanebadCveulebriv drekadi wonasworobis iseT amocanebs gulisxmoben,roca sxeulis zedapirze ganlagebulia sxvadasxvagvari tipissasazRvro pirobebis gamyofi xazebi. Tu gansaxilveli drekadisxeulis zedapiri Sedgeba ramdenime gluvi waxnagisagan, maSinSereuli amocanis xarisxobrivad gansxvavebuli ori ZiriTadivarianti SeiZleba warmogvidges.1) TiToeuli waxnagis farglebSi sasazRvro pirobebis tipiar icvleba. aseTi Sereuli amocanebis umartives magaliTebswarmoadgenen drekadi fenis wonasworoba, romlis erT waxnagzemocemulia Zabvebi, xolo meoreze gadaadgilebebi da aseveanalogiuri amocanebi solisaTvis, Rru cilindrisaTvis,konusisaTvis da sxva. miTiTebuli konkretuli amocanebisamonaxsnebi SeiZleba miviRoT furies, hankelis da msgavsiintegraluri gardaqmnebis gamoyenebiT. rogorc miuTiTes g.popovma da n.rostovcevma (1966) [88], amgvari tipis zogadi64

problemebi principSi daiyvaneba gantolebaTa usasrulosistemebamde.2) sxeulis erT waxnagze mainc gvaqvs sxvadasxvagvari tipissasazRvro pirobebis gamyofi xazi. aseTi tipis problemebi,romlebic, zogadad Tu vityviT, daiyvaneba integralurgantolebebamde, Cven gvinda aq gavarCioT ufro detalurad,radgan saxeldobr maT misces biZgi, potencialis Teoriisa dadrekadobis Teoriis bevri mniSvnelovani Sereuli amocanisamoxsnis sxvadasxvagvari meTodebis ganviTarebas. amasTan erTadmsgavs Sereul amocanebs miekuTvneba rigi gamoyenebiTisakiTxebisa da kerZod, kontaqturi amocanebi da zogierTi amocanaZabvebis koncentraciis Sesaxeb.dReisaTvis zedmiwevniT detalurad Seswavliliadeformirebuli wriuli an elifsuri xisti tvifariT, drekadinaxevarsivrcis sakontrolo amocanebi. pirvelad msgavsi amocanaganixileboda jer kidev J. busineskis mier wriuli cilindrisuxaxunod RerZuli Sewnevis SemTxvevisaTvis. amocanaTa amkategorias miekuTvneba h. hercis klasikuri problema drekadtanTa kumSvis Sesaxeb iseT pirobebSi, roca sakontaqto farTobielifsi aRmoCndeba. sakiTxTa am wris Semdgom ganviTarebaSierTob arsebiTi wvlili Seitanes sabWoTa mecnierebma. a. dinikma(1909) da n. bediaevma (1924) Caatares Zabvebis gamoTvla sxeulebSi,romlebic erTmaneTs exebian wriul an elifsur farTobze (ix.agreTve m. kroleveci (1966). sakontaqto amocanebSi sakmaod didiraodenobis mniSvnelovani samuSaoebi Sesrulda ocdaaTian daormocian wlebSi. v. abramovma (1939) da a. luriem (1940) mogvcessakontaqto amocanebis amonaxsni aracentralurad datvirTulimrgvali da elifsuri tvifaris Sesaxeb. am mimarTulebiTarsebiTi Sedegebi miiRo i. Staermanma (1939, 1941, 1943), romelmacganixila brunvis sxeulebis kontaqtis sxvadasxva SemTxvevebi,maTi TanaSexebis zedapiris simciris Sesaxeb daSvebis gareSe, daagreTve gamoikvlia pirvelad amocana tvifaris mkvrivadmibrjenis Sesaxeb. 1941 wels a. luriem lames funqciis65

saSualebiT dawvrilebiT ganixila zogierTi sakontaqtoamocana, amasTan SeimuSava bunebrivi da erTsaxovani midgomahercis amocanisadmi da mkvrivad mibrjenis Sesaxeb. m. leonovisa(1939, 1940) [89] da l. galinis (1946, 1947) naSromebSi mocemulianaxevarsivrcisaTvis rigi sakontaqto amocanebis Semdgomiganzogadeba. originaluri da mimoxilviTi xasiaTis didi masala,romelic exeba ganxilul problemebs, Sedis i. Staermanis (1949)[90], l. galinis (1953) [91], a. luries (1955) monografiebSi daagreTve d. Sermanisa (1950) da g.Sapiros (1950) mimoxilviTstatiebSi.momdevno wlebSi drekadobis Teoriis zogadi gantolebebisgamoyenebaze da kerZod, papkoviC-neiberis funqciis gamoyenebazedafuZnebuli meTodebis ganviTarebam, SesaZlebloba mogvcanaxevarsivrcis drekadi wonasworobis bevri saerTo Sereuliamocana dagveyvana potencialis Teoriis Sereuli amocanebiszogierT klasebamde. amasTan am amocanebidan mizanSewoniliagamovyoT is SemTxveva, roca naxevarsivrcis mTel sazRvarzemocemulia mxebi Zabvebi, romeliRac sasrul midamoSi SsasazRvro sibrtyisa z=0. cnobilia normaluri gadaadgilebau z= f ( x,y), xolo S-is gareT (S′ midamoSi) mocemulia normaluriZabva σ = σ ( x,y). ase, sakontaqto amocanisaTvis xaxunis dazdatvirTvebis gareSe gvaqvs σ≡0, xolo funqcia f ganisazRvrebatvifaris fuZis formiT. arsebiTia, rom miTiTebuli klasisSereuli amocanebi saboloo angariSSi SeiZleba davides erTiharmoniuli funqciis moZebnamde, romelic mocemulia S midamoSi,amasTan S′ midamoSi cnobilia misi normaluri warmoebuli.potencialis Teoriis msgavsi amocanebis meTodurma damuSavebamsaSualeba mogvca zustad amogvexsna zogierTi sakontaqto damsgavsi Sereuli amocanebi. am meTodebidan ZiriTadebi arian:sferoidaluri da elifsoidaluri koordinatebis gamoyeneba(a.lurie); grinis funqciis ageba da gamoyeneba (l. galini,m.leonovi, 1953); integralur gantolebaTa meTodi (i. Staermani,66

v.mosakovski, 1953); toroiduli koordinatebisa da integralurigardaqmnebis gamoyeneba (i.ufliandi 1956, 1967) [92]; kompleqsuripotencialebis meTodi (m.rostovcevi, 1953, 1957). Cven aqspecialurad ar gamovyofT wyvil integralur gantolebaTameTods, romelic warmatebiT ganaviTara i.sneginma, ramdenadacmisi efeqturoba arsebiTad gamJRavndeba ufro rTuli Sereuliamocanebis gadawyvetisas, rasac qvemoT SevexebiT.dasaxelebulma da agreTve bevrma sxva avtorma ukanaskneliaTwleulebis ganmavlobaSi mogvces sivrciTi drekadobisTeoriis axali Sereuli amocanebis amomwuravi amonaxsnebi, maTricxvSi kontaqturisac. ase, l.galinma (1947) da v. rvaCiovma (1949)[93] ganixiles naxevarsivrceSi solisebri tvifaris CawnevissakiTxi; n. kilCevskis (1958, 1960) naSromebSi mocemulia hercisamocanis ganzogadeba da miTiTebulia drekadi kontaqtisamocanis romeliRac eqstremalur problemasTan kavSiri;v.rvaCiovma (1956, 1957) gadawyvita amocanebi tvifaris Sesaxebzolisa da mravalkuTxedis saxiT, agreTve tvifaris SemTxveva,roca mas fuZe meore rigis mrudiT aqvs SemosazRvruli. g.popovis naSromebi miZRvnilia Sereuli amocanebisadmi kontaqtiswriuli midamosaTvis da tvifarisaTvis naxevarsibrtisa dakvadrantis saxiT; n.borodaCevma (1962, 1964, 1966). a. xrustaliovma(1965) gamoikvlies rigi Termodrekadi amocanebisanaxevarsivrcisaTvis. gansakuTrebiT sayuradReboa amocananaxevarsivrceze Rru wriuli cilindris moqmedebis Sesaxeb,romelic literaturaSi cnobilia rgoluri Stampis amocanissaxeliT. am amocanis zusti amoxsna dakavSirebulia ovaluri kveTisrgolis aratabulirebul funqciebTan (ix. n. lebedevi, 1937). igiveamocanis amoxsnis sxva saxis miaxloebiTi meTodebiSemoTavazebulia a.aleqsandrovis (1955), i. arkadievis (1962), v.gubenkos da v. mosakovskis (1960) [94], k. egorovis (1963), g. popovis(1967) mier. ukanasknel wlebSi daisaxa am amocanisadmi da masTanmimsgavsebuli amocanebisadmi, romlebic dafuZnebulia wyviliintegraluri gantolebebis gamoyenebaze da dakavSirebulia67

eler-fokis gardaqmnebTan (v.grinCenko da a. ulitko, 1963; a.babloiani, 1964; a.ruxoveci da i. ufliandi 1965-1967), da agreTvesammagi integraluri gantolebebis gamoyenebaze (n. borodaCevi daf. borodaCiova, 1966), midgomis kidev erTi gza. miTiTebulimeTodebi saSualebas gvaZleven miviRoT efeqturi miaxloebebi,romlebic dafuZnebulia fredholmis integralur gantolebaTaricxviT amonaxsnebze.gamoqveynebulia Sereuli amocanebisadmi miZRvnilinaSromTa didi raodenoba, romlebic dakavSirebulia drekadfuZeze mdebare koWebisa da filebis Runvis sakiTxebTan.gamovyofdiT a. iSkovas (1947), m. leonovis (1939) da v. pakmovis(1960) gamokvlevebs, romlebic exeba drekad naxevarsivrcezemrgvali filis Runvas, da aseve m. gorbunova-posadovis (1953) dab. korenevis (1954, 1960) [95] monografiebs. am mimarTulebis bevrinaSromis Sedegebsa da did bibliografias mkiTxveli ipovis a.iSkovasa da b. korenevis (1966) mimoxilviT moxsenebaSi.kontaqtur amocanebTan erTad, zemoT ganxilulipotencialis Teoriis Sereuli amocanebi naxevarsivrcisaTvisSeiZleba iqnan ganmartebulni, rogorc SemousazRvreli drekaditanis, romelic dasustebulia brtyeli xvretebiT, rasac ukavia Smidamo (an S′), deformaciis Sesaxeb amocanebi. marTlac xvretebisnapirebis CatvirTvis SemTxvevaSi, simetriuliT xvretebissibrtyis mimarT, sakmarisia ganvixiloT naxevarsibrtye, romlissazRvarze S midamoSi (an S′-Si) mocemulia Zabvebi, xolo misgareT mxebi Zabvebi da normaluri gadaadgilebebi aRar arian.antisimetriuli CatvirTvis SemTxvevaSi wriuli xvritisaTvisacki warmoiqmneba zogierTi damatebiTi siZnele, gadawyvetili v.mosakovskisa (1955) da i. ufliandis (1967) naSromebSi, amasTanukanasknel naSromSi es amocana ganxilulia rogorc zogadiSereuli amocanis kerZo SemTxveva, roca naxevarsivrcis mTelsazRvarze normaluri Zabvaa S(S′) midamoSi cnobili mxebigadaadgileba, xolo S′(S) midamoSi mocemulia mxebi Zabvebi. ori68

sxvadasxva garemos kontaqtis Sesaxeb saintereso amocana,romelTa saerTo sazRvarze aris mrgvali xvreti, gadawyvetiliav. mosakovskisa da m. ribkas (1964) mier; am dros xorcieldebagrifit-snedonis cnobili kriteriumis ganzogadeba araerTgvarovanisxeulis SemTxvevaze (ix. agreTve amave avtorebis statia, 1965).xvretebis mqone sxeulebis deformaciebTan dakavSirebulnaSromTagan mivuTiTebT kidev v. grinCenkos da a. ulitkos (1965),v. aleqsandrovisa da b. smetaninis (1965) [96] sainteresostatiebze da agreTve i. ufliandis (1958) naSromze, romeliceZRvneba naxevradusasrulo brtyeli Wrilis mqone taniswonasworobis Sesaxeb amocanas.Cvens mier ganxilul kontaqtur amocanebTan dakavSirebulnaSromTa umetesobaSi igulisxmeboda, rom xaxuni tvifarsa dadrekad tans Soris ar arsebobda. gacilebiT did maTematikursirTuleebs warmoadgens meore zRvruli mdgomareoba, rocatvifari da fuZe imyofeba SeWidulobis pirobebSi (aseTi amocanaaris drekadobis Teoriis ZiriTadi Sereuli amocanebis kerZoSemTxveva). ufro martiv SemTxvevaSi saqme daiyvanebanaxevarsivrceSi ori harmoniuli funqciis moZebnaze pirveli dameore rigis ganuyofeli sasazRvro pirobebiT. pirvelad aseTiamocana wriuli StampisaTvis iqna gadawyvetili v. mosakovskis(1954) mier, ori analizuri funqciis wrfivi SeuRlebis brtyelamocanaze dayvanis gziT. SemdgomSi i. ufliandma (1965, 1967)mogvca am amocanis uSualo amoxsna toroidaluri koordinatebisada meler-fokis integraluri gardaqmnebis saSualebiT. b.abramianis, n. aruTunianisa da a. babloianis (1966) statiaSiganxorcielebulia kidev erTi midgoma amave amocanisadmi,romelic dafuZnebulia wyvil integralur gantolebaTagamoyenebaze. kontaqtur amocanebs, SeWidulobis arsebobisas,eZRvneba agreTve v. mosakovskis (1963) naSromi. drekadobis TeoriisZiriTadi Sereuli amocanis amoxsna naxevarsivrcisaTvis,romelsac sasazRvro pirobebis sworxazovani gamyofi sazRvari69

aqvs, mogvca i. ufliandma (1957) kontoroviC-lebedevis integralurigardaqmnis saSualebiT.tvifaris sasazRvro xazis siaxloves ZabvaTa qcevisgamokvlevas, roca igi imyofeba SeWidulobis pirobebSi, eZRvnebag. savinisa [97] da v. rvaCiovis (1963) statia.drekad naxevarsivrceSi xisti tvifaris Cawnevis Sesaxebklasikuri amocanis bunebriv ganzogadebas warmoadgenssakontaqto amocana SemousazRvreli drekadi fenisaTvis amsakiTxTa kvleva intensiurad tardeboda ormocdaaTian wlebSi,amasTan naxevarsivrcis SemTxvevisagan gansxvavebiT, aq ukve veRarxerxdeboda zusti amonaxsnebis miReba, da SeiZleboda mxolodSesabamisi amocanis dayvana integralur gantolebebze. pirvelnaSromad aq unda miviCnioT b. koganis statia (1954), romelSicSedgenilia da ricxobrivad gadawyvetilia pirveli gvarisintegraluri gantoleba mrgval tvifarsa da fenas Sorissakontaqto wnevisaTvis. msgavsi amocanis ufro efeqturi amonaxsnimogvca n. lebedevma da i. ufliandma (1958) [98], romlebicganixilavdnen gegmaSi mrgvali, xisti tvifaris RerZul Cawnevasdrekad fenaSi, romelic ido xist fuZeze, xaxunis ararsebobisas. es amocana daiyvaneboda wyvil integralurgantolebaze∞∫0∞∫0⎫Φ(λ)I0( λr)dλ= f ( r)(0 ≤ r < a)⎪⎬dλλΦ(λ)I= < < ∞ ⎪0( λr)0, ( a r )1−g(λ)⎪⎭−λhλh+ eg( λ)=,λh+ chλhshλhsadac a _ tvifaris radiusia, h _ fenis sisqe, f(r) _ mocemulifunqcia, romelic dakavSirebulia tvifaris fuZis formasTan,Φ(λ) _ saZiebeli sidide. axali ucnobi funqciisagan kvadraturissaxeSi amonaxsnis warmodgenis gziT[ 1−g(λ)] ∫Φ( λ ) = ϕ(t)cosλtdt.meore gantoleba kmayofildeba igivurad, xolo pirvelidaiyvaneba fredholmis gantolebaze, romelsac uwyvetia070

simetruli guli aqvs. amoxsnis aseTi gza SesaZleblobas iZlevaCavataroT rigi ricxviTi gaangariSebebi, kerZod, vipovoT Stampisgadaadgilebasa da P RerZul Zalas Soris damokidebulebamartivi formuliTa∫P = 2π ϕ( t)dt .0k. egorovma (1960) gamoiyena msgavsi meTodika tvifarisaraRerZuli Cawnevis SemTxvevaze v. pupirevisa da i. ufliandisstatiaSi (1960) da ukanasknelis monografiaSi (1967) mocemuliaamonaxsni Sereuli amocanis drekadi fenisaTvis, da agreTveganxilulia fenasa da fuZes Soris SeWidulobis SemTxveva.arsebiTia mivuTiToT, rom wyvil integralur gantolebaTameTodma saSualeba mogvca efeqturad gangvexila ufro rTuliRerZulsimetriuli amocanac sxvadasxva radiusis ori tvifariTfenis kumSvis Sesaxeb (i. kuzmini da i. ufliandi, 1967). i.voroviCma da i.ustinovma (1959) miiRes singularuli integralurigantoleba uSualod Φ(λ) funqciisaTvis da SeimuSaves misiamoxsnis miaxloebiTi meTodi a/h-s xarisxebad mwkrivSi gaSlisgziT. analogiuri meTodi iyo gamoyenebuli d. grilickis miermravalfeniani garemos grexis Sesaxeb amocanaSi masTan SeWidulitvifaris saSualebiT da msgavs kontaqtur amocanebSi. wyvilintegralur gantolebaTa meTodma saSualeba misca rig avtorebs(magaliTad. g. valovi, 1914, s. kotliari, 1964; v. devnoroviCi, 1964)[99] amoexsnaT sxvadasxvagvari sakontaqto amocanebi drekadifenisaTvis, maT ricxvSi Termodrekadic. sakontaqto da Sereuliamocanebi anizotropuli tanebisaTvis ganixileboda s. lexnickis(1950), d. grilickisa da i.kizimas (1962, 1964) [100] , r. sunCeleevis(1965, 1966) mier.specialuri efeqturi meTodi tvifaris drekad fenazezemoqmedebis Sesaxeb sakontaqto amocanebisadmi, dayrdnobiliintegralur gantolebis uSualo ganxilvaze, tvifaris qveSwnevisaTvis, iyo SemoTavazebuli v.aleqsandrovisa da i.voroviCis (1960, 1964) mier. amocanis amonaxsns hqonda mcire71

parametriT gaSlis saxe – tvifaris maxasiaTebeli zomisSefardeba fenis sisqesTan. arsebiTia, rom efeqturi Sedegebi amdros moxerxda ara marto wriuli, aramed gegmaSi elifsuriStampisaTvis, da aseve fuZis zogierTi sxva formebisTvisac.miTiTebulma meTodma miiRo Semdgomi ganviTareba v. aleqsandrovisa(1963, 1964, 1967) [101] da i. voroviCis naSromebSi. igi dReisaTvisSeiZleba CaiTvalos erTerT yvelaze efeqtur meTodadgansaxilveli klasis sakontaqto amocanebis amoxsnisaTvis, fenissisqisa da tvifaris maxasiaTebeli zomis fardobis nebismiersididisas.naSromTagan, romlebic eZRvneba ufro rTul sakontaqtoproblemebs, aRvniSnoT v. gubenkos statia (1960), romelSicgamokvleulia rgoluri tvifarebis drekad fuZeze zemoqmedeba daagreTve i.voroviCisa da v.kopasenkos (1966) naSromi kontaqturiamocanis Sesaxeb naxevarzolisaTvis.wyvil integralur gantolebaTa saSualebiT, SeiZlebawarmatebiT gadawydes amocanebi drekad fenaSi Zabvebiskoncentraciis Sesaxeb, romelic dasustebulia TanaRerZulixvretebiT, paralelurebiT fenis sazRvrebisa. aseTi tipisumartives amocanas (i. ufliandi, 1959) warmoadgens drekadi feniswonasworoba, romelic Sua sibrtyeSi Seicavs erT simetriuladCatvirTul wriul xvrets. i.markuzovma (1963) gamoikvlia igivesakiTxi wonasworuli bzaris zomebis povnis Sesaxeb amocanasTankavSirSi g.bareblatis meTodiT.sxva naSromebisagan, romlebic exeba Wrilebisa daxvrelebis mqone sxeulebis wonasworobas, mivuTiTebT v.panasiukis (1960) [102], n. lebedevis da i. ufliandis (1960), i.kuzminisa (1966) da n. palcunis (1967), b.mixailovis (1974), g. yifianis(1989) [103]. statiebsa da agreTve g. savinis, a.kosmodamianskis da a.guzis (1967), n. preobraJenskis (1989), b.mixailovisa da g. yifianis(1988), a. filinis (1987) mimoxilviT naSromebs.gadavideT axla kontaqtur amocanebze, romlebic exebausasrulo cilindris wonasworobas. am sakiTxebis ganxilvisas72

yvelaze efeqturi aRmoCnda wyvil integralur gantolebaTameTodi, dakavSirebuli furies gardaqmnebTan RerZulikoordinatiT. am xerxis damaxasiaTebel Taviseburebas warmoadgensis garemoeba, rom kontaqtis naxevradusasrulo midamosSemTxvevaSi es gantolebebi kompleqsuri cvladis funqciisTeoriis meTodebis saSualebiT, romlebic mocemul zolSianalizuri funqciis faqtorizaciis SesaZleblobas eyrdnobian,uSveben zust amonaxsns. am mimarTulebis pirvel naSromad iqca b.koganis (1956) statia, miZRvnili usasrulo cilindris, romelicCaWedilia uxaxunod naxevradusasrulo xist garsakrSiRerZulsimetriuli daZabuli mdgomareobis SeswavlisadmidaSvebiT, rom kontaqtis midamoSi mocemulia mudmivi radialurigadaadgileba, amocana daiyvaneba wyvil gantolebaze, saxiT∞∫0∞∫0f ( λ)eiλξdλ= 0,2 iλξf ( λ)I1( λ)e dλ22 2( λ − 2 + 2v)I ( λ)+ λ I1= u0,( λ)20⎫( ξ > 0), ⎪⎬( ξ < 0) ⎪⎪⎭romelTa zusti amoxsna xorcieldeba raime melomorfulifunqciis usasrulo namravlis saxiT agebiT. b. koganis, a.xrustaliovis da f.veinSteinis (1958-1965) ufro gviandelnaSromebSi es meTodika iyo gamoyenebuli sxvadasxva SereulamocanebSi, rogorc mTliani, aseve Rru cilindrisaTvisac daagreTve transversaluri anizotropiis SemTxvevaSic. msgavsiamocanebis amoxsnis meTodi, dafuZnebuli maTi viner-hopfisintegralur gantolebaze dayvanaze kontaqturi ZabvebisaTvis,SeimuSava g.popovma (1964) [104]. mis miervea mocemuli kontaqtisori simetriuli ubnis mqone usasrulo cilindrisaTvissakontaqto amocanis amoxsna. mivuTiTebT kidev g. valovis (1966)statiaze, sadac trigonometriuli guliT wyvili integralurigantolebebis saSualebiT ganxilulia Rru usasrulocilindris grexis Sesaxeb amocana.bolo periodSi, wyvil mwkrivTa axali aparatis ganviTarebisSedegad moxerxda amoxsnadi sakontaqto amocanebis arsebiTi73

gafarToeba. Sereuli amocanebisadmi mimarTebaSi drekadisferosaTvis wyvil mwkrivTa qveS (an wyvil jamTa gantolebebisqveS) igulisxmeba gantolebaTa sistema∞∑n=0∞∑n=0α A K ( x)= fnγ A K ( x)= fnnnnn10( x),( x),⎫( a ≤ x < c),⎪⎬( c < x < b),⎪⎪⎭romlisganac unda ganisazRvros A n koeficientebi, amasTanigulisxmeba, rom gulebi K n (x) qmnian (a, b) SualedSi Sekrulsistemas, xolo α n da γ n mocemulia wyvil mwkrivTa saSualebiT,romlebic Seicaven gaSlas leJandris polinomebad; n.aruTinianis,b. abramianis da a. babloianis (1964, 1966) naSromebSi amoxsniliqna ramdenime saintereso amocana drekadi sferos deformaciisSesaxeb, da agreTve brunvis elifsoidis Sesaxeb, SereulisasazRvro pirobebisas. maT mier ganxilulia sferosRerZulsimetriuli kumSva ori simetriulad ganlagebuli,erTnairi xisti tvifariT, xaxunis ar arsebobis daSvebiT. esamocana moxerxda dayvaniliyo zemoT miTiTebuli saxis wyvil1mwkrivze, rac Kn( x)= Pn( x),αn= n + , γn=1 + βn( βnsidideebs n → ∞ -2isas aqvT 1/n rigi), a =_1, b =1. Tu aRvniSnavT V(x)-iT wyvilmwkrivTagan pirveli jamis mniSvnelobas, roca x>c, maSin amoxsnadaiyvaneba integralur gantolebazesadacx11 d dyV ( x)=ξπ dx x − y∫ ∫V( ξ ) S(ξ,y)d = Φcc( x),⎡⎛⎞ ⎤= ∑ ∞ 1S(ξ , y)2 βnPn( ξ )cos⎢⎜n+ ⎟arccosy⎥n=0 ⎣⎝n ⎠ ⎦da Φ(x) _ cnobili funqciaa. msgavsi meTodiTaa mocemulidrekadi sfero masTan SeWiduli da simetriulad ganlagebulierTnairi ori tvifariT, grexis amocanis amonaxsni. leJandrispolinomebad wyvil mwkrivTa meTodiT mocemulia agreTvezogierTi Sereuli amocana drekadi sferos da gawelili74

unvis elifsoidis kumSvisa da grexis Sesaxeb. da bolos,ganxilulia sakontaqto amocana drekad sferoSi xisti tvifarisCawnevisa. amasTan wyvili mwkrivebi leJandris polinimebaddayvanil iqna wrfiv algebrul gantolebaTa usasrulo sistemaze.magaliTis saxiT ganixileboda sfero, romelic uZravadaauxaxunod naxevarsferul amonaRebSi da datvirTulia zedapirisdanarCen nawilSi.wyvili mwkrivebi leJandris polinomebad SeiZleba iyosaseve efeqturad gamoyenebuli sferuli koordinatebis saSualebiTsferuli CanarTis mqone naxevarsivrcis grexis Sereuli amocanisamoxsnisas (ix. a. ruxoveci da i. ufliandi, 1967).aRsaniSnavia agreTve n. borodaCevis (1967) saintereso statia,romelSic wyvili mwkrivebi beselis funqciebad gamoyenebulianaxevradusasrulo cilindris torsSi wriuli tvifaris CawnevisSesaxeb RerZulsimetriul amocanaSi.saWiroa mivuTiToT drekadobis Teoriis sivrciTi Sereuliamocanebis kidev erT nawilze, romelmac ukanasknel wlebSimiiRo didi ganviTareba. kerZod, sakontaqto amocanebi wrfivaddeformirebadi fuZisaTvis da masTan dakavSirebul amocanebze,araerTgvarovan drekad naxevarsivrceze tvifaris zemoqmedebisSesaxeb. fuZemdebeli naSromebi aq ekuTvnis b. korenevs (1954, 1957,1960). SemdgomSi am problemebiT iyvnen dakavebuli v. mosakovski(1958), g. popovi (1959), a. rakovi da v. rvaCiovi (1961), n. rostovcevi(1961, 1964) da rigi sxva avtorebisa. ufro dawvrilebiTi cnobebiam sakiTxebze Sedis b. iSkovasa da b. korenevis mimoxilviTmoxsenebaSi (1966).dasasrul aRvniSnoT, rom mniSvnelovani raodenoba cnobebisada didi bibliografia drekadobis Teoriis Sereuli sivrciTiamocanebis Sesaxeb, romlebic Seswavlilia meoce saukunis 60-ianiwlebis CaTvliT, moipoveba d. Sermanis (1962), g.abramianis da a.aleqsandrovis (1966), g. popovisa da n.rostovcevis (1966), n.kilCevskisa da e. kostiukis (1966), v.rvaCiovis (1967) mimoxilvebSi.75

2.2.1. drekadobis Teoriis brtyeli amocanebis dasmada amoxsnis meTodebidReisaTvis erTerTi yvelaze mniSvnelovani da kargaddamuSavebuli drekadobis Teoriis nawili, sadac miRwevebigansakuTrebulad didia, aris e.w. drekadobis brtyeli Teoria.brtyeli amocanebis damuSavebaSi warmateba aixsneba misiganxilvisas kompleqsuri cvladis analizuri funqciis TeoriisgamoyenebiT. pirveli Sedegebi am mimarTulebiT, riTacmTlianobaSi ganisazRvra brtyeli Teoriis Tanamedrove saxe,miRebuli iyo g. kolosovisa da n.musxeliSvilis fuZemdeblurgamokvlevebSi.drekadobis Teoriis brtyeli amocana gulisxmobs drekadigaremos deformacias, romelic mocemuli sibrtyis (grZelicilindris deformacia Tavisufali fuZeebiT) paraleluria, anmis brtyel daZabul mdgomareobas (Txeli firfitis deformaciaZalebiT, romlebic mdebareobs mis sibrtyeSi). am SemTxvevebSidrekadi wonasworobis gansazRvra daiyvaneba biharmoniuligantolebisaTvis sasazRvro amocanebis amoxsnaze. biharmoniulgantolebamde dadis, normaluri datvirTvebis qveS moqceulidrekadi firfitebis, wonasworobis amocanebic. maTematikuriformulirebiT brtyeli amocanebi da firfitebis Runvisamocanebi erTmaneTis msgavsia, aseve msgavsia. maTi amoxsnismeTodebic. amitom mizanSewonilia am ori tipis amocaniserToblivi ganxilva.2.2.1.1. brtyeli amocanis amoxsnis zogadi kompleqsuriwarmodgenan. musxeliSvilis aRniSvnebSi nagulisxmevia, rom brtyeliamocanis ZiriTadi Tanafardobebi cnobilia. drekadi garemos Sare, warmoadgens Oxy sibrtyis bmul nawils, SemosazRvrulserTi an ramdenime Sekruli konturiT L1 , L2,L , L m, L m + 1saerTo76

wertilebis gareSe. amasTan, ukanaskneli moicavs yvela danarCens.Lm+1konturis ar arsebobisas gvaqvs naxvretebiani sibrtyisusasrulo are. ganixileba agreTve SemTxveebi, roca L k konturebsSoris gvaqvs sasruli sigrZis an usasrulo Seukreli nawilebi(sibrtye bzarebiT, naxevarsibrtye naxvretebiT da a.S.)igulisxmeba masaTa Zalebis arqona.Zabvebi da deformaciebi kolosov-musxeliSvilis ϕ(z), ψ(z)kompleqsur potencialebSi gamoisaxeba formulebiTXYyx+ Y− Xyx= 2+ 2iX[ ϕ′( z)+ ϕ′( z)],y= 2[ zϕ′′ψ ( z)+ ψ ′(z)],2μ(u + iv)= xϕ(x)− 2ϕ′(z)−ψ( z).am formulebze pirvelad miuTiTa g. kolosovma 1909 welsfuZemdeblur naSromSi `kompleqsuri cvladis funqciis TeoriiserTi gamoyenebis Sesaxeb drekadobis maTematikuri Teoriisbrtyel amocanaSi~. maTi mkacri dasabuTeba mocemuliamogvianebiT n. musxeliSvilis mier (ix. misi monografia`drekadobis maTematikuri Teoriis zogierTi ZiriTadi amocana~,1933, gam. 5-1966) [105].n. musxeliSvilis kalams ekuTvnis 80-mde samecnierogamokvleva. misma naSromebma didi gavlena iqonies drekadobisTeoriisa da maTematikuri fizikis Semdgom ganviTarebaze mTelmsoflioSi. n. musxeliSvilis meTodebis safuZvelze daweriliauamravi naSromi. misi Sedegebi farTodaa gamoyenebuli agreTveteqnikis sxvadasxva amocanebSi. n. musxeliSvilis fundamenturimonografia `drekadobis maTematikuri Teoriis zogierTiZiriTadi amocana~, romelic pirvelad 1933 wels gamosca ssrkavSiris mecnierebaTa akademiam, Semdgom kidev xuTjer daibeWda.misi monografiebi gadaTargmnilia ucxo enebze. ar SeiZleba araRiniSnos, rom Targmanebis gamoqveynebas ucxoel mecnierTawreebSi aRfrTovanebiT Sexvdnen. ucxoeTis samecniero dareferatuli Jurnalebis furclebze gamoqveynda recenziebi,romlebSic monografias Zalze maRali Sefaseba aqvs micemuli.⎫⎪⎬⎪⎭(61)77

qvemoT moyvanilia nawyvetebi cnobili ingliseli meqanikosisprofesor r. hilis recenziidan, romelic inglisur JurnalSi`Nature~ (t. 174, # 4433, 1954w.) gamoqveynda saTauriT: `etapidrekadobis TeoriaSi~.am Targmanis gamoCena, mxedvelobaSia zemoT aRniSnulimonografia, romelic yovelmxriv Cinebulia, pirvelxarisxovanimniSvnelobis movlenas warmoadgens. musxeliSvilma aCvena, romzogad amoxsnebs (gadaadgilebaTa da daZabulobaTa veliskompleqsur warmodgenebs) aqvs ara marto Teoriuli mniSvneloba,aramed sagrZnobi upiratesoba rig sasazRvro amocanebSisayovelTaod miRebul meTodebTan SedarebiT.`musxeliSvilis nawarmoebebma ar SeiZleba uaRresad RrmaSTabeWdileba ar moaxdinos mkiTxvelze Tavisi mkafiobiT,sasiamovno darbaisluri stiliTa da yovelgvari detalisadmikeTilsindisieri damokidebulebiT, arsad ar gvxvdebadauxveweloba (naSromebi ki Seicaven 1000 gverdze mets), yoveliarsebiTi adgili mkafiodaa axsnili: msjeloba umaRles donezeaayvanili damatebebSidac ki. dasasruls recenzenti aRniSnavs:`samwuxaroa, rom avtoris TvalTaxedva aris TvalTaxedvamaTematikosisa, romelic meqanikaSi muSaobs. sakiTxi araa mTelisigrZe-siganiT ganxiluli: gaSuqebulia erTi aspeqti, magram rabrwyinvaled!~ analogiuri Sefaseba gvxvdeba mraval sxvarecenziebSic.aRsaniSnavia, rom caladbmul da sasrul S areSi, SeyursuliZalebisa da momentebis ar qonisas, potencialebi ϕ(z) da ψ(z)holomorfulia. mravalbmuli sasruli ares SemTxvevaSi kiZabvebisa da gadaadgilebebis calsaxobisa da sasrulobismoTxovnas S-Si mivyavarT warmodgenebamde1ϕ(z)= −2π(1 + x)1ψ ( z)= −2π(1 + x)m∑k = 1m∑k = 1( Xk( Xk+ iYk− iYk)ln( z − Z)ln( z − Zkk* ⎫) + ϕ ( z),⎪⎬*) + ψ ( z),⎪⎪⎭(62)78

sadac ϕ * (z) da ψ * (z) holomorfulebia S-Si, Zk− Lk-s SigniTwertilebia,X + iY L -ze gare Zalebis mTavari veqtoria.kkkusasrulo S aresaTvis, roca Lm+ 1konturi ar gvaqvs, xoloZabvebi sxeulis usasrulod daSorebul nawilebSi sasrulia, ϕda ψ warmodgenebs usasrulod daSorebuli wertilis siaxloveseqnebaT saxeX + iY⎫ϕ(z)= − ln z + ϕ0( z)+ Γz,2π(1 + x)⎪⎬X − iYψ ( z)= x ln z + ψ ( ) + Γ′ . ⎪0zz2π(1 + x)⎪⎭Γ, Γ′ kompleqsuri mudmivebiT ganisazRvreba Zabvebi dagadaadgilebebi usasrulobaSi, X + iY − gare Zalebis mTavariveqtoria L ares srul sazRvarze, xolo ϕ 0 (z) da ψ 0 (z)holomorfulebia z = ∞ midamoSi. gadaadgilebis veqtoriSemosazRvrulia Γ = Γ′ = 0 , X + iY = 0 pirobebisas.(63)2.2.2. drekadobis Teoriis ZiriTadi brtyeliamocanebis formulirebadrekadobis brtyeli Teoriis ZiriTad amocanebadCveulebriv gulisxmoben Semdeg sams:pirveli ZiriTadi amocana moiTxovs sxeulis drekadiwonasworobis gansazRvras, roca mis sazRvarze mocemulia gareZalebi. es amocana daiyvaneba analizuri funqciis Teoriis SemdegsasazRvro amocanamde:ϕ ( t ) + tϕ′( t)+ ψ ( t)= f ( t)+ c(t), L-ze (64)sadac f(t)_L-ze mocemuli funqciaa, romelic ganisazRvreba gareZalebiT formuliTsf ( t)= i∫ ( Xn+ iY0n) ds,amasTan s arisL kkonturis rkali. igi aTvlilia TiToeulL k-ze79

misi romeliRac fiqsirebuli wertilidan dadebiTi mimarTulebiT,xoloC =( t)CklL -ze,C k-kompleqsuri mudmivia.meore ZiriTadi amocanaa sxeulis drekadi wonasworobisgansazRvra misi sazRvris wertilebis mocemuligadaadgilebebiT. am SemTxvevaSi S areSi ϕ da ψ analizurifunqciebis moZebnisaTvis unda gvqondes sasazRvro pirobax ϕ ( t)− tϕ′( t)−ψ( t)= g(t), L-ze (65)sadac g(t) mocemuli funqciaa, g ( t)= 2μ ( u + iv)L-ze.ZiriTad Sereul amocanas simartivisaTvis CamovayalibebTsasruli caladbmuli aresaTvis, romelic SemosazRvrulia erTiSekruli konturiT. am amocanaSi sazRvris nawilzeL ′ = a1 b1+ a2b2+ L + a nb n, sadac a kbk( k = 1, L,n)-is garkveuli wesiTganlagebuli konturis aragadamfaravi rkalebia, mocemuliagare Zabvebi, xolo L′ = b1a2+ b2a3+ L + b na n+ 1( a n + 1= a1)sxva nawilzemocemulia gadaadgilebebi. analizuri funqciebis Sesabamisamocanas aqvs saxek ϕ ( t)+ tϕ′( t)+ ψ ( t)= h(t)+ c(t). (66)(64) da (65) pirobebi daculi unda iyos TiToeul Lkkonturze. c(t) mudmivas, romelic figurirebs (64)-is marjvenanawilSi SeuZlia miiRos sxvadasxva mniSvneloba sxvadasxvakonturze. mxolod erT maTganze SegviZlia igi davafiqsiroTsurvilisamebr (Cveulebriv miiCneven, rom cm+1= 0 ), xolo sxvakonturebze isini sruliad nebismieri rCebian da gansazRvraseqvemdebareba amocanis gadawyvetis msvlelobaSi. sruliad aseveckmudmivebi (66)-is marjvena nawilSi (garda Cvens mierSerCeulisa) isini araa winaswar mocemuli da eqvemdebarebiangansazRvras ϕ da ψ funqciebTan erTad.drekadobis brtyel TeoriaSi ganixileba agreTve e.w. mesameZiriTadi amocana, roca garemos sazRvarze mocemuliagadaadgilebis veqtoris normaluri mdgeneli da gare Zabvebisveqtoris mxebi mdgeneli. es Seesabameba drekadi tanis mocemuli80

formis xist profilTan urTierTSexebas, roca drekad da xistsxeulebs Soris kontaqti xdeba mTel maT sazRvarze.Tu nebismieri mudmiva (64) da (66)-is marjvena nawilebSiisea dafiqsirebuli, rogorc miTiTebulia zemoT, maSin ϕ da ψ-saTvis damatebiT pirobebs eqnebaT Semdegi saxe:pirvel amocanaSiϕ ( 0) = 0, Im ϕ ′(0)= 0 .meore da Sereul amocanebSiϕ (0) = 0 an ψ ( 0) = 0 .amiT amoiwureba ϕ da ψ-is amorCevis neba.damtkicebulia, rom dasmul amocanaTagan TiToeuls arSeiZleba qondes erTi amonaxsnis meti. pirveli ZiriTadiamocanis amoxsnis arsebobisaTvis ares sazRvarze modebuli gareZalvebis mTavari veqtorisa da mTavari momentis nulTantolobis pirobebia aucilebeli. (64)-is marjvena nawilSifigurirebuli f(t) funqciis calsaxobisa da uwyvetobisas, es oripiroba daiyvaneba erTze (n. musxeliSvili, 1966).Re ∫ f ( t)dt = 0 .Lfirfitebis Runvis TeoriaSi mtkicdeba, rom TxelierTgvarovani drekadi firfitis, romelic moqceulia miszedapirze ganawilebuli normaluri datvirTvis moqmedebis qveS,Sua sibrtyis CaRunva ω(x, y) akmayofilebs araerTgvarovanbiharmoniul gantolebasqΔΔω = , (67)Dsadac q _ datvirTvis intensiobaa, xolo D _ cilindrulisixiste.(67)-is raime kerZo amonaxsnis moZebnis Semdeg, Cven SegviZliagursas cnobili formuliT warmovadginoT am gantolebiszogadi amonaxsni or analizur ϕ da x funqciaSi, amasTanχ ′( z)= ψ ( z). maTi meSveobiT gamoisaxeba ZiriTadi sidideebi,romlebic gansazRvraven firfitis daZabul mdgomareobas.81

adgili aqvs Semdeg formulebs (s. lexnicki, 1938), romlebickolosov-musxeliSvilis formulebis analogiuria:aqMx,MMNxyx− M+ M− iNyxy+ 2iHxy= −8D(1− v)= 4D(1− v)= −8Dϕ′′( z)+ N[ zϕ′′ ( z)+ ψ ′(z)][ ϕ′( z)+ ϕ′( z)]0x0− iN .y+ M0x+ M+ M0y,0y− M0x+ 2iH9xy, ⎫⎪⎬⎪⎪⎭(68)M − mRunavi momentebia, H − mgrexi momenti, N , −yxyxN ygadamWreli Zalebi, romlebic modian sigrZis erTeulze;0 0M L,N − igive sidideebi, romlebic miekuTvneba (67) gantolebisx,yamorCeul kerZo amonaxsns. ϕ da ψ funqciebis gansazRvrulobisxarisxi igivea, rac brtyel amocanaSi.(67) gantolebidan CaRunvis gansazRvrisaTvis aucilebelia masmivuerToT sasazRvro pirobebi, romlebic Seesabamebian sazRvrisCamagrebis ama Tu im xasiaTs.aq gvaqvs Semdegi sami ZiriTadi amocana. CamovayaliboTisini Sekruli konturiT SemosazRvruli caladbmuli garemosaTvis.I. firfitis kide Camagrebulia _ niSnavs, rom firfitis Suazedapiris mier dakavebuli S aris sazRvarze unda gvqondesSemdegi Tanafardobebidωω =0, = 0,dnsadac n _ konturisadmi gare normalia.saxe(69)II. firfitis kide Tavisufalia _ sasazRvro pirobebs aqvT222⎡∂ω⎫2 ∂ ω 2 ∂ ω ⎤vΔω+ (1 − v)⎢ cos θ + sin θ + sin 2θ⎥ = 0,22⎪⎣ ∂x∂y∂x∂y⎦ ⎪⎬2 22∂Δω1−v d ⎡⎛∂ ω ∂ ω ⎞ ∂ ω ⎤+=⎪⎢⎜ −⎟sin 2θ+ 2 cos2θ⎥ 0,2 2∂n2 ds⎪⎣⎝∂y∂x⎠ ∂x∂y⎦ ⎭sadac θ _ kuTxea, romelsac adgens gare normali Ox RerZTan,tolobaTa marcxena nawilebi warmoadgenen Sesabamisad mRunavmomentsa da ganzogadebul gadamWrel Zalas, mikuTvnebulebssigrZis erTeulisadmi da moqmedebs firfitis elementze nnormaliT.(70)82

III. firfitis kide dayrdnobilia _ kidis Tavisufaldayrdnobas pasuxobs Semdegi pirobebi:ω ≡ 0,2⎡∂ωvΔω+ (1 − v)⎢ cos2⎣ ∂x22∂ ωθ + sin2∂y2⎫2⎪∂ ω ⎤ ⎬θ + sin 2θ⎥ = 0.∂x∂y⎪⎦ ⎭garda sasazRvro pirobebis am ZiriTadi saxeebisa,araiSviaTad gvxvdeba gamoyenebisaTvis gansakuTrebiT saintereso,Sereuli pirobebi, roca, magaliTad sazRvris erTi nawiliCamagrebulia, sxva dayrdnobili, da danarCeni Tavisufali.vinaidan w funqciisa da misi normaluri warmoebulissasazRvro mniSvnelobiT yovelTvis SeiZleba moiZebnos x da y-iTam funqciis kerZo warmoebulebis sasazRvro mniSvnelobebi,amocana erTi firfitis Runvis Sesaxeb brtyeli drekadobisTeoriis pirveli amocanis tolfasia. pirveli amocanissasazRvro pirobebi zustad emTxveva (64) pirobas.Tavisufal sazRvarze (70) pirobas, rogorc es iyoaRniSnuli s.lexnickis (1938) da i. vekuas (1942) naSromebSi, maTisaTanado gardaqmnebiT daiyvaneba kompleqsuri cvladis funqciisTeoriis sasazRvro amocanamde, romelic (65)-is analogiuria.gansxvaveba mdgomareobs mxolod imaSi, rom mudmiva x (65)-is+marcxena nawilSi icvleba sxva mudmivaTi: x = ( 3+v)(1− v), xolomarjvena nawili moicema ict + c1saxis mamravlis sizustiT, sadacc _ nivTieri, c 1-kompleqsuri mudmivaa. Tumca, caladbmuli aresgansaxilvel SemTxvevaSi es mudmivebi SeiZleba miviCnioTYnulad.da bolos kideebis Tavisufali dayrdnobis (71) pirobebiSeiZleba ϕ da ψ funqciebiT Caiweros saxiT (a. kalandia, 1953)(71)⎪⎧⎛Re⎨λ′0ϕ( t)− ⎜⎪⎩ ⎝⎧ dtRe⎨⎩dsdtds⎞⎟⎠2[ tϕ′′( t)+ ψ ′(t)]⎪⎫⎫⎬ = g1(t),⎪⎪⎭ ⎪⎬⎫[ ϕ(t)+ tϕ′( t)+ ψ ( t)] ⎬ = g ( ), ⎪ ⎪ 2t⎭⎭L-ze (72)sadac g 1,g2− L -ze mocemuli funqciebia, λ0= 2(1 + v) /(1 − v).83

ar aris Zneli davrwmundeT, rom (72) amocana da drekadobisbrtyeli Teoriis mesame amocana erTmaneTis tolfasia.zemoT naTqvamidan naTelia, rom brtyeli amocanebis amoxsnismeTodebi, xandaxan, TiTqmis yovelgvari cvlilebis gareSeudgeba Txeli firfitebis Runvis amocanebs. es SesaZleblobapirvelad gamoyenebuli iqna a. luries mier (1928).2.2.3. brtyeli amocanebis amoxsnis meTodebiqvemoT mocemulia brtyeli amocanebis amoxsnis mokledaxasiaTeba, romlebic efuZneba kompleqsuri cvladis funqciisTeoriis gamoyenebas. ZiriTadad Cven SemovifarglebT imSemTxvevis ganxilviT, roca drekadi garemo avsebs caladbmulsasrul ares, SemosazRvruls Sekruli konturiT. L-is mimarTSida ZiriTadi S ares, aq+S -iT aRvniSnavT, xolo gares(romelic S + -s Seavsebs srul sibrtyemde) _S − -iT.2.2.3.1. funqciis holomorfuloba (analizuroba)gavixsenoT zogierTi elementaruli cneba da daSveba,romlebic ukavSirdeba analizuri funqciis Teorias dagamoiyeneba Semdgomi msjelobisas.koSis integralis qveS igulisxmeba gamosaxuleba1 f ( t)dtF(z)= ∫ , (73)2πiL t − zsadac t _ L konturis wertilis afiqsia, xolo z _ sibrtyisnebismieri wertilis afiqsi. Tu z _ emTxveva L konturis t 0wertils, maSin (73) integralis qveS koSiT, misi mTavarimniSvneloba gaigeba.F(z) funqcia, romelic (73) formuliTaa gansazRvruli,holomorfulia rogorc+S areSi, aseve−S -Sic f(t)-s sakmarisisigluvisas (magaliTad Tu igi akmayofilebdes helderis pirobas84

L-ze) uwyvetia Sesabamis Sekrul+S +Lda−S + LareebSi. amfunqciis zRvruli mniSvnelobebi L-dan marcxniv da marjvniv,romeliRact 0∈ L wertilSi, aRniSnuli Sesabamisad F + ( t 0)daF − ( t 0)moicema soxocki-plemelis cnobili formulebiT.rogorc+S -Si. aseve−S -Si holomorful da+F dauwyveti zRvruli mniSvnelobebis mqone funqcias n.musxeliSvilis mixedviT, uban-uban holomorfuls uwodeben. ubanubanholomorfuli funqciis magaliTs, f(t)-s funqciis mimarTcnobili pirobebisas iZleva (73) integrali.imisaTvis, rom mocemuli, L-ze uwyveti f(t) funqcia iyosromelic f(z) funqciis zRvruli mniSvneloba, romelicholomorfulia S + -Si, aucilebeli da sakmarisia piroba1 f ( t)dt∫ = 0 yvela z ∈ S− − isTvis. (74)2πiL t − zanalogiurad, imis piroba, rom funqcia f(t) iyos zRvrulimniSvneloba f(z) funqciis, romelic holomorfuliatoloba1 f ( t)dt∫ = const2πi L t − zim SemTxvevaSi, rocayvela−FS − -Si, arisz ∈ S+ − saTvis. (75)+S warmoadgens wres erTeuliradiusiT, winamdebare pirobebs SeiZleba mivceT ramdenadme sxva,Semdgomi miznebisaTvis ufro moxerxebuli saxe (n. musxeliSvili,1966). mocemuli f(z) funqciiT, romelic holomorfulia+S -Si,ganvsazRvroT kompleqsuri argumentis sxva funqcia SemdegitolobiT⎛ 1 ⎞f* ( z)= f ⎜ ⎟ . (76)⎝ z ⎠xandaxan am funqciisaTvis Cven gamoviyenebT aRniSvnas⎛ 1 ⎞f* ( z)= f ⎜ ⎟ . (77)⎝ z ⎠koSi-rimanis pirobis uSualo SemowmebiT adviladdavrwmundebiT, rom f ( ) funqcia holomorfulia −S* zareSi,85

usasrulod daSorebuli wertilis CaTvliT da piriqiT. Tufunqcia f(z) holomorfuliaS − -Si, maSin holomorfuli iqneba z-sagan S + areSi.(76) aRniSvna SeiZleba gamoviyenoT ufro zogadSemTxvevaSic, roca, magaliTad, f(z)-s S + -is SigniT aqvs polusebissasruli ricxvi. f ( ) funqcias maSin eqneba imave rigis* zpolusebi wertilebSi, romlebic f(z)-is polusebis asaxvebiaerTeulovan wrewirSi.(76) funqciis zRvruli mniSvnelobebisaTvis gveqneba++−f ( t ) = f ( ) , f ( t)= f ( ) . (78)*t*t(74)-is gamoyenebiT f ( ) funqciis mimarT, miviRebT pirobas* z1 f ( t)dt+∫ = const yvela z ∈ S -saTvis. (79)2πiL t − zaucilebels da sakmariss imisaTvis, rom wrewirze uwyveti f(t)funqcia iyos romeliRac iseTi f(z) funqciis sasazRvromniSvneloba, romelic holomorfulia S + -Si. mudmivas, romelictolobis marjvena nawilSi figurirebs, aqvs gansazRvrulimniSvneloba f (0). analogiurad, (75) piroba miiRebs saxes1 f ( t)dt∫ = 02πit − zLyvela−z ∈ S -saTvis. (80)(76) operacia iZleva S + -Si holomorfuli, mocemuli f(z)-iT−S -Si, holomorfuli funqciis konstruirebis erT SesaZlebelxerxs. cxadia, rom wreSi holomorfuli funqciis gavrcelebaSeiZleba ganxorcieldes xerxebis uricxvi simravliT. saxeldobr,gavrcelebis miTiTebebuli xerxi aris erTi im mcireTagan,romlis gamoyeneba SedarebiT iolia.f(z) funqcia, romelic gansazRvrulia rogorc S + -Si, ise−S -Si formuliT⎪⎧f ( z),rocaf ( z)= ⎨⎪⎩ f ( z),roca*zz < 1> 1.86

aSkaraa, rom f(z) uban-uban holomorfulia. garda amisa, igianalizurad gangrZobadia t = 1 wrewiris im ubnebiT, romlebzecIm f ( t)= 0 . f(z)-is ukanaskneli Tviseba uSualod gamomdinareobs(78)-dan.holomorfuli funqciebis amgvari gavrcoba xSirad+gamoiyeneba dasaxuli miznebiT im SemTxvevaSic, roca S -naxevarsibrtyea. maSin (76)-is nacvlad miiCneven (n. musxeliSvili,1966).F ( z)= F(z). (81)brtyeli amocanebis amoxsnis mravalnairi meTodebidan,romlebic cnobilia samecniero literaturaSi dReisaTvis, aqSevexebiT ZiriTadad imaT, romlebic yvelaze ufro efeqturebiarogorc sasazRvro amocanebis zogadi gamokvlevebis azriT, isekerZo SemTxvevebSi konkretulad maTi SeswavlisaTvis.upirvelesad yovlisa, mxedvelobaSi gveqneba Semdegi oTximeTodi.1. xarisxovani mwkrivebis meTodi konformuli asaxvisgamoyenebiT;2. dayvana funqcionalur (kerZod, integralur)gantolebebamde konformuli asaxvis gamoyenebiT (erTbmuliareebis SemTxveva);3. zogadi meTodebi, romlebsac mivyavarT integralurgantolebebamde konformuli asaxvis gareSe;4. dayvana wrfivi SeuRlebis amocanamde.rig specialur SemTxvevebSi, gansakuTrebiT mravladbmulgaremoTa Seswavlisas, mizanSewonilad miiCneva ganxilvisas amaTu im meTodTa urTierTSexamebiT Semotana.warmovadginoT miTiTebuli meTodebis mokle aRwera.87

2.2.3.2. mocemuli funqciis konformuli asaxvabrtyeli amocanis amoxsnisas xSirad sasargebloa winaswarkonformulad avsaxoT mocemuli are, romelic Sevsebuliadrekadi garemoTi, sibrtyis romeliRac sxva areze damxmare ζcvladiT. sasruli caladbmuli S aresaTvis, romelicSemosazRvrulia Sekruli konturiT, Cveulebriv mimarTavenerTeuli radiusis wreze asaxvas sasruli oradbmuli midamosaTvis– wriul koncentrirebul rgolze. naxevarusasrulo aresSemTxvevaSi, romelsac orive mxares usasruli sami mimavalisazRvari aqvs – naxevarsibrtyeze da a.S.Cven aq mivuTiTebT konformuli asaxvis gamoyenebis erTvariantze miTiTebul SemTxvevaTagan pirvelSi (n. musxeliSvili,1956). vTqvaTz = ω(ζ ) ,aris Tanafardoba, romelic axdens erTeuli wris ζ < 1, romliskonturic Cven aRvniSneT γ-Ti saWiro konformuli asaxvisrealizebas S areze. funqciebs ϕ(z) da ψ (z), romlebic axali ξcvladiTaa gamosaxuli aRvniSnavT ϕ(ζ), ψ(ζ).sadacfunqciaa.pirveli amocanis (64) sasazRvro pirobebi miiReben saxesω(σ )ϕ( σ ) + ϕ′( σ ) + ψ ( σ ) = f ( σ ) γ-ze, (82)ω′( σ )σ − γ konturis wertilia,σ = ei∂ , f −γ-ze mocemulidavuSvaT, furies kompleqsuri mwkrivebis saxiT gaSlebisSemdegi SesaZlebloba.ω(σ )ω′( σ )= ∑ ∞−∞kb σ ,k∑ ∞ A k−∞kf ( σ ) = σ , (83)da davuSvaT, rom erTeul wreSi ( ζ < 1-isas)ϕ(ζ ) =ψ ( ζ ) =∞∑0∞∑1ka ζ , ψ ( ζ ) =kka ζk.k−1∞∑0⎫a′′kζ,⎪⎬⎪⎭(84)88

maSin (82)-is safuZvelze, winare mwkrivebis krebadobis mimarTcnobili pirobebisas, Cven mivdivarT ucnobi koeficientebis a k,a′kgansazRvrisTvis gantolebaTa Semdeg sistemebamdeam+ ∑ ∞ ka ==k 1∞a′ m+ ∑kak=1kbm+k−1 Am= 1,2, )( m L , (85)kb− m+k−1 = A−m( = 0,1,2, L)m . (86)damtkicebulia, rom (85) wrfiv gantolebaTa usasrulosistema gadawyvetadia, Tu daculia statikis pirobebi, da rommisi amonaxsni (86)-Tan erTad iZleva gansaxilveli brtyeliamocanis amonaxsns mocemuli f(t) funqciis sakmarisi sigluvisas.praqtikisaTvis gansakuTrebuli mniSvneloba aqvs Semdegfaqts. im SemTxvevaSi, roca amsaxveli funqcia warmoadgenspolinomsω = L +2n( ζ ) c1ζ+ c2ζ+ c nζ ( 1≠ 0, cn≠ 0)c . (87)(85) usasrulo sistema gadagvardeba Semdeg sasrul sistemadam= Am( m ≥ n + 1)⎫a⎪1+ a1b1+ 2a2b2+ L+nanbn= A1⎪a2+ a1b2+ 2a2b3+ L+( n −1)an−1bn= A2, ⎬................................................................... ⎪⎪an+ a1bn= An.⎪⎭xolo (86) formula gvaZlevs(88)m∑ + n + 1k=1a ′ + ka b A ( m = 0,1,2, L). (89)mk − m+k−1=amocana daiyvaneba (88) sasruli sistemis amoxsnaze.miTiTebuli meTodi, aSkaraa, gamosadegia wriul rgolzeasaxvis SemTxvevaSic.zogierTi konkretuli amocanis amoxsnisas xarisxovanimwkrivebis meTodi konformul asaxvasTan erTad (SeerTebaSi)farTod gamoiyeneba dRemde. is xandaxan gamoiyeneba ramdenadmeSecvlili saxiTac (ix. magaliTad, d. Sermani, 1951, k. grei, Quart J.Mech. and Appl. Mech., 1953, 1954).−m89

2.2.3.3. koSis tipis integralebis gamoyenebaamocanis efeqturi amoxsnisaTvis gansakuTrebiT sasargebloaRmoCnda qvemoT gadmocemuli meTodi, romelic uxamebskonformul asaxvas koSis tipis integralebis aparatisgamoyenebas (n. musxeliSvili, 1966, $$ 78-85). is mdgomareobsSemdegSi.(82) sasazRvro pirobebidan gamomdinare da pirobisgamosaxvidan, rom ψ(σ) warmoadgens sasazRvro mniSvnelobas ψ(ζ)funqciis wrewirze, romelic holomorfulia wris SigniT danuli xdeba roca ζ=0, miviRebT (79)-is safuZvelze, funqcionalurgantolebas1 ω(σ ) ϕ′( σ )ϕ( ζ ) + ∫ dσ= A(ξ ) , ( ζ ∈ γ ) , (90)2πiω′( σ ) ( σ −ζ)γ1 f ( σ ) dσA( ζ ) = ∫ .2πiσ −ζdamtkicebulia, rom fiqsirebuli Im[ ϕ (0) ω′(0)]γ′ mudmivisas,(90) gantoleba calsaxad gansazRvravs ϕ(ζ) funqcias. misigansazRvris Semdeg ψ(ζ) funqcia moiZebneba uSualod (82)-dankoSis integralis saSualebiT.(90) funqcionaluri gantoleba saSualebas gvaZlevssruliad elementaruli saSualebebiT avagoT amocanis zustiamonaxsnebi areebis farTo klasisaTvis. miaxloebiTi amonaxsniki SeiZleba principSi miviRoT caladbmuli ares yvelaze zogadiSemTxvevisaTvis.naTqvamis sailustraciod ganvixiloT SemTxveva, rocaamsaxveli ω(ζ) funqcia racionaluria. am SemTxvevaSigamosaxulebaω(σ )ϕ′( σ ) ,ω′( σ )romelic integralis niSnis qveS figurirebs (90)-Si, warmoadgenssasazRvro mniSvnelobas funqciisa90

ω(ζ ) ⎛ 1 ⎞ϕ ′ ⎜ ⎟ , (91)⎛ 1 ⎞′ ⎜ ⎟⎝ ζω⎠⎝ ζ ⎠romelic holomorfulia γ-is gareT, polusebis sasruli ricxvis_ ω(ζ) funqciis gansakuTrebuli wertilebis gamoklebiT.radgan (90) integralSi wertili ζ Zevs γ-s SigniT, esintegrali gamoiTvleba sasruli saxiT da warmoadgensracionalur funqcias, romelic Seicavs gaSlis ucnobikoeficientebis raRac ricxvs.aqedan gamomdinareobs, swored, n. musxeliSvilis cnobiliwinadadeba, romlis Tanaxmadac brtyeli amocanis amoxsna,areebis gansaxilveli klasisaTvis, SeiZleba miviRoTkvadraturebSi, wrfiv algebrul gantolebaTa sasruli sistemisamoxsnamde sizustiT. kerZod SemTxvevaSi ϕ(ζ) funqciis (87) saxispolinomuri asaxva (90)-Si, warmogvidgeba, koSis A(ζ)integralebsa da ζ-Ti n xarisxis polinoms, romelic Seicavsucnobis koeficientebis saxis TviT ϕ(ζ)-is pirvel n koeficients,jamis saxiT. gantolebaTa wrfivi sistema, miRebuli amukanasknelTa gansazRvrisaTvis, zustad emTxveva (88) sistemas.orive saZiebeli funqcia ϕ(ζ) da ψ(ζ), ganisazRvreba CaletilisaxiT am sistemis amonaxsniTa da mocemuli f funqciiT.Tu funqcia ω(ζ) araracionaluria, magram misi wreSi gaSlacnobilia, meTods mivyavarT wrfiv gantolebaTa usasrulosistemamde, es ki iZleva saSualebas avagoT amocanis amonaxsninebismieri mocemuli sizustiT.2.2.3.4. mravladbmul areebSi kompleqsuri warmodgenaϕ′ (σ ) drekadi velebis kompleqsuri warmodgena analizurifunqciebis sxvadasxvagvar integralur warmodgenebTan naerTSiwarmoadgens brtyeli amocanis integralur gantolebebamdedayvanisaTvis moxerxebul aparats. dReisaTvis cnobilia aseTi91

gantolebebis agebis ramdenime varianti. mivuTiToT zogierTmaTganze.ϕ′ (σ ) mimarT fredholmis integraluri gantoleba SeiZlebamiviRoT (3.30) funqcionaluri gantolebidan, Tu mas winaswarCavwerT ramdenadme sxva saxiT da mere mivaswrafebT ξ wertilsSignidan γ wrewiris wertilisaken (n. musxeliSvili, 1966, $ 79), amintegraluri gantolebis elementaruli gamokvleva saSualebasiZleva davamtkicoT misi amonaxsnis arseboba (maSasadame, Sesabamisibrtyeli amocanis amoxsnis arsebobac), Tu, ra Tqma unda, pirveliamocanis SemTxvevaSi sasruli garemosaTvis daculia statikispirobebi. ufro dawvrilebiT am gantolebis gamokvleva Caatara d.Sermanma (1938). man Seiswavla integraluri gantolebismaxasiaTebeli wertilebis ganawileba da daamtkica, rom igiamoxsnadia orive ZiriTadi amocanisaTvis mimdevrobis miaxloebismeTodiT.mravladbmuli areebis SemTxvevis momcvel, ufrozogad meTods warmoadgens integralur gantolebebamde miyvanismeTodi, romelic winaswar ar Txoulobs konformul asaxvas.erT-erTi aseTi meTodi iyo wamoyenebuli n. musxeliSvilis (1966,$ 98) mier. mis arss Cven ganvmartavT daSvebiT, rom garemosasrulo da caladbmulia.(64) gantolebaSi, romelic gamosaxavs amocanis sasazRvropirobas, gadavideT SeuRlebul mniSvnelobaze da (74)-is TanaxmadCavweroT imis piroba, rom funqcia ψ(t)_z-iT sasazRvromniSvnelobaa funqciis, romelic holomorfuliamiviRebT maSin funqcionalur gantolebas+S -Si.Cven1 ϕ(t)dt 1 tϕ′( t)dt2πit − z 2πt − z∫ + ∫ =LLA(z)yvela−z ∈ S -sTvis,A(z)=12πi∫Lf ( t)dt.t − zTu Cven gamovsaxavT im pirobas ϕ(t) da ϕ′ (t)-saTvis, maSingveqneba ori sxva toloba, romelic analogiuri iqneba winasi. am92

sami tolobis kombinirebiT, maTSi z-iT marjvnidan zRvarzewinaswari gadasvlis Semdeg miviRebT fredholmis gantolebas ϕ(t)-saTvis, romelic miTiTebuli aqvs n.musxeliSvils.1 t − t01 t − t0ϕ ( t0) + ∫ϕ(t)d ln + ∫ϕ(t)d = −A(t0). (92)2πit − t 2πit − tL0winamdebaresTan erTob msgavsi, magram arsiT gansxvavebulibrtyeli amocanis gantoleba aago d. Sermanma (1940) sxva gziT,razedac Cven ufro dawvrilebiT qvemoT aRvniSnavT:d. Sermanis (1935-1937) gamokvlevebis Tanaxmad, (92) gantolebavargisia nebismieri mravladbmuli aresaTvis; mas yovelTvis aqvsamonaxsni, romelic iZleva Sesabamisi brtyeli amocanisamonaxsns. garda amisa odnav winaswar Secvlili (92)gantolebisaTvis mimdevrobiTi miaxloebis meTodia gamosadegi(d. Sermani, 1940).brtyeli amocanis integraluri gantoleba, aseve vargisinebismieri mravladbmuli aresaTvis, iyo agebuli ufro adre s.mixlinis (1934, 1935) mier. am mizniT ganxilvaSi Semodis e.w.grinis kompleqsuri funqcia, xolo Semdeg misi saSualebiT –Svarcis ganzogadebuli guli, analizuri areSi, magram aracalsaxad. mravladbmul areSi ganzogadebul guls gaaCnia Tviseba,analogiuri Svarcis Cveulebrivi gulis Tvisebisa. mixlinisgantoleba caladbmuli aresaTvis Tanxvdeba (92) gantolebas. s.mixlinma Caatara agebuli gantolebebis gamokvleva;damtkicebuli iqna maTi amoxsnadoba da agreTve, maTiamoxsnisaTvis mimdevrobiTi miaxloebis meTodis samarTlianoba.Sedegebi gadmocemulia mis monografiaSi (1949), romelic SeicavsagreTve, brtyeli amocanis amoxsnisaTvis Svarcis gulisgamoyenebas zogierT kerZo SemTxvevaSi.l. maRnaraZis gamokvlevebma (1937, 1938) gviCvenes, rommusxeliSvilis gantoleba inarCunebs Zalas texili sazRvriskuTxeebis SemTxvevaSic, Tu integralebs, romlebsac gantolebaSeicavs, gavigebT garkveulad ufro ganzogadebuli azriT.L093

mravladbmuli ares zogad SemTxvevaSi integralurigantolebis martivi da mravalmxriv moxerxebuli forma iyonapovni 1940 wels d. Sermanis mier. moviyvanoT Sermanisgantolebis gamoyvana, SemovifargloT ra, uwindeburad,caladbmuli sasruli ares SemTxveviT. amjerad, pirveli dameore ZiriTadi amocanebi ganvixiloT erTdroulad,gavaerTianebT ra sasazRvro pirobebs Semdeg tolobaSi:k ϕ ( t)+ tϕ′( t)+ ψ ( t)= f ( t)L-ze (93)sadac k = 1-s pirveli amocanis SemTxvevaSi da k = −χ-s meoreSemTxvevaSi. Sermanidan gamomdinare, miviCnioT S areSiϕ(z)=12πikψ ( z)=2πi∫Lω(t)dt,t − z∫Lω(t)dt−t − z12πi∫L⎫⎪⎬tω′( t)dt, ⎪t − z ⎪⎭sadac ω(t)_L konturis wertilis raime funqciaa, romeliceqvemdebareba gansazRvras, Tu am formulebSi gadavalT zRvarze,roca z wertili Signidan miiswrafis wertilisaken t 0∈ L , daCavsvamT (93)-Si napovn sasazRvro pirobebs, maSin zogierTimartivi gardaqmnis Semdeg miviRebT Tanafardobas(94)1 t − t01 t − t0k ω ( t0 ) + ∫ω(t)d ln − ∫ ω(t)d = f ( t0) . (95)2πit − t 2πit − tL0es fredholmis integraluri gantolebaa ω t ) -saTvis,romlis Sesaxebac zemoT gvqonda laparaki. is atarebslauriCela-Sermanis gantolebis saxels.mravladbmuli garemos SemTxvevaSi mizanSewonilia, Sermanidangamomdinare, (94) warmodgenas ramdenadme saxe vucvaloT, risSedegadac Seicvleba (95) gantolebac.gamokvleva gviCvenebs, rom laureCela-Sermanis erTgvarovangantolebas ar gaaCnia aratrivialuri amonaxsnebi da miserTaderT amonaxsns iZleva sawyisi sasazRvro amocana (94)formulebiT.L0( 094

(94) warmodgena gamosadegia agreTve ZiriTadi Sereuliamocanis amoxsnisaTvis. Tumca am SemTxvevaSi Cven saqme gveqnebaintegralur gantolebebTan, romlebic Seicaven koSis tipisgulebs, romelTa Teoria damuSavebulia dReisaTvis iseTivesisruliT, rogorc fredholmis gantolebebisaTvis (n.musxeliSvili, 1946, 1952; n. vekua, 1950).integraluri gantolebebi uTuod warmoadgenen sasazRvropirobebis zogadi gamokvlevebisaTvis moxerxebul saSualebas,kerZod maTi amonaxsnis arsebobis damtkicebisaTvis. Tumca, amasTanerTad, integralur gantolebaTa meTods Tvlian naklebefeqturadda arc Tu usafuZvlod. am meTodis safuZvelze amocanebispraqtikuli amoxsnis cdebi, misi Cveuli sqemiT, romelic iyenebsgaTvlebisaTvis integraluri gantolebebis diskretul analogs,naklebad manugeSebelia Tanamedrove gamoTvliTi saSualebebiTacki. amitom mravladbmuli garemos zogadi SemTxvevisaTvis,amoxsnis ufro metad misaRebi algoriTmis ar qonis gamo, gviwevsamoxsnebis, romlebic amocanebis ama Tu im klass moergeba,efeqturi specialuri meTodebis Zieba.am azriT did mniSvnelobas iZens zemoT CamoTvlilimeTodebis sxvadasxva kombinaciebi. Cven mxedvelobaSi gvaqvs,upirveles yovlisa, funqcionaluri gantolebebis Sexamebaxarisxovani mwkrivebis meTodTan, funqciaTa wrfivi SeuRlebismeTodisa konformul asaxvasTan, da agreTve amoxsnis ufrozogadi sqemebi, romlebic gzadagza integralur gantolebaTaaparats iyeneben. am specialur xerxTagan zogierTi qvemoT iqnebamiTiTebuli.2.2.3.5. brtyeli amocanis amoxsnis xerxebid. Sermanis (1947, 1951) rig SromebSi SemuSavebuli iqnabrtyeli amocanis amoxsnis efeqturi xerxi (sasruli dausasrulo) oradbmuli areebis gansazRvruli klasebisaTvis,romlebic SemosazRvrulia ori Sekruli mrudiT. meTodis95

ZiriTad niSnad, romelic dasaSvebi areebis klass gansazRvravs,iTvleba moTxovna, rom brtyeli amocana caladbmuli aresaTvis(ares SemomsazRvrel Sekrul konturTagan erTis mimarT gare anSiga) uSvebs amonaxsns Caketili saxiT. amrigad, ares sazRvradSeiZleba gamodiodnen wrewirebi, elifsebi, wesierimravalwaxnagebi momrgvalebuli wveroebiT da a.S. usasruloares magaliTia _ moTxovnili saxis ori naxvretis mqone sibrtyeganxilvaSi SeiZleba CavrToT naxevarsibrte ori naxvretiT(samadbmuli are, Tu CavTvliT, rom naxvretebi misi sworxazovanisazRvridan mdebareoben Sors da am ukanasknelze moiTxovebasasazRvro pirobebis Sesruleba mxolod miaxloebulad). amtipis amocanebi gansakuTrebiT mniSvnelovania samTo saqmeSigamoyenebisaTvis. meTodis arsis gadmocemisas CavTvaloTgarkveulobisaTvis S are sasrulod, romelic SemosazRvruliaL 1 (Sida) da L 2 (gare) mrudeebiT.SemovitanoT ganxilvaSi damxmare ω(t) funqcia, romelicgansazRvrulia L 2 -ze, Tanaxmad tolobisaϕ ( t)− tϕ′( t)−ψ( t)= 2ω(t)(t L 2 -zea) (96)Tu jer SevkrebT, xolo mere gamovaklebT erTmaneTs wevrwevrad(96)-sa da (94)-s, CavTvliT ra rom C = 20 , miviRebT,ϕ(t)= ω(t)+ f ( t),ψ ( t)= −⎬[ ] ⎫ω(t)+ tω′( t)+ [ f ( t)− tf ′(t)].⎭(97)ω(t)-s saSualebiT SemoviRoT ori axali ϕ 0 (z) da ψ 0 (z)funqcia, Semdegi saxissadac1 ω(t)dt⎫ϕ0( z)= ϕ(z)− ∫ − F(z),2πi−⎪L t z2⎬1 ω(t)+ tω′( t)ψ ( ) = ( ) +− ⎪0z ψ z ∫dt G(z),2πit − z⎪L2⎭F(z)=12πi∫L2f ( t)dt 1 f ( t)− tf ′(t), G(z)= ∫ dt .t − z 2πit − zL2(98)Tu ufro gamovkveTavT saZiebel ϕ da ψ funqciebs,CavTvliT, ra maT L 2 -is gareT nulis tolad, maSin (97) toloba,96

ogorc amaSi advilad davrwmundebiT gaxdeba L 2 konturs iqiT,axlad SemoRebuli funqciebis analizurad gagrZelebadobispiroba. am, yvelgan garda L 1 -sa, holomorfuli ϕ0da ψ0funqciebisaTvis (64) tolobis safuZvelze L 1-ze gveqneba sasazRvropiroba[ t;ω()]ϕ0 ( t)+ tϕ′( t)+ ψ0( t)= Ω t , (99)sadac Ω _ raRac wrfivi operatoria.meTodis ZiriTadi moTxovnis Tanaxmad Cven miviCnevT, rom(99) damxmare brtyeli amocana gadawyvetadia sasrulo saxiT.aSkarad es yovelTvis iqneba saxeldobr ase, Tu funqcia,romelic asaxavs ares wreze L 1 -is gareT, racionaluria.Ω marjvena nawiliT, romelic droebiT iTvleba t-ganmocemul funqciad, moiZebneba musxeliSvilis funqcionalurgantolebaTa meTodiT; (99) amocanis amoxsna Caketili saxiT danapovni ϕ0, ψ0funqciebi Seitaneba (96) pirobaSi. es iZleva ω(t)-sgansazRvrisaTvis Tanafardobas fredholmis meore gvarisintegraluri gantolebis saxiT. Semdeg gamoiyeneba ω(t)-s gaSlafuries kompleqsur mwkrivebSi da integraluri gantolebadaiyvaneba wrfiv algebrul gantolebaTa usasrulo sistemaze.2.2.3.6. lauriCela-Sermanis gantolebis gamoyenebarig SemTxvevebSi integraluri gantolebebi SeiZlebauSualod gamoyenebul iqnan amocanaTa efeqturi amoxsnisaTvis. CvenmivuTiTebT lauriCela-Sermanis gantolebis gamoyenebis erTSesaZleblobaze.davuSvaT, rom CvenTvis cnobilia ω(ζ) funqcia, romelicaxdens wris konformul asaxvas areze (L konturis mimarT anSida an gareze). Tu (95) gantolebaSi movaxdenT cvladis Secvlast = ω(σ ) tolobis Tanaxmad, maSin miviRebT integralurgantolebas erTeuli radiusis wrexazze. am gantolebis guli97

elementarulad gamoisaxeba ω(σ)-Ti da inarCunebs martivstruqturas bevr SemTxvevaSi. magaliTad (87) saxis nebismieriasaxvis SemTxvevaSi. yvela am SemTxvevaSi axlad miRebuliintegraluri gantolebisaTvis gamoiyeneba furies mwkrivebismeTodi, rasac mivyavarT amocanis efeqtur amoxsnamde.2.2.3.7. helderis pirobis gamoyenebawrfivi SeuRlebis sasazRvro amocanis qveS CvenvigulisxmebT Semdeg amocanas: moiZebnos F(z) funqcia, romelicholomorfulia L xazis gaswvriv gaWril kompleqsur sibrtyeze,sasazRvro pirobiT+−F ( t)= a(t)F ( t)+ b(t), (100)sadac a(t) da b(t)_L-ze mocemuli funqciebia, F + (t)da F − (t)_L-zesaZiebeli F(z) funqciis sasazRvro mniSvnelobebia, L xazisgaswvriv arCeuli dadebiTi mimarTulebis mimarT, marjvniv damarcxniv. igulisxmeba, ra es sasazRvro pirobebi arseboben yvelgan,garda, SesaZloa, L xazis C1, C2,L , Cm, wertilebs sasruloricxvisa, romelTa modamoSi F(z) gvaZleven SefasebasAF( z)≤ (A da α-mudmivebia, α < 1).αz − C kxandaxan iZebneba (100) sasazRvro amocanis amonaxsnebi,romlebic dauSvebs poluss L-ze aramdebare, sibrtyis romelimewertilSi. Cveulebrivad aseT wertilad miiReba usasruloddaSorebuli wertili.Cven ganvixilavT (100) amocanas Semdegi daSvebebisas: LSedgeba sasrulo ricxvis aragadamkveTi Sekruli an Seukvreligluvi konturebisagan, a(t) da b(t) funqciebi akmayofilebenhelderis pirobas L-ze, garda pirveli gvaris wyvetiswertilebis sasruli ricxvisa, a ( t)≠ 0 . am daSvebebSi (100)amoixsneba cxadi saxiT (kvadraturebSi). amoxsnas (romelsac aqvspolusi usasrulobaSi) aqvs saxe98

X ( z)b(t)dtF( z)= ∫ + X ( z)P(z), (101)+2πiL X ( t)(t − z)sadac P(z)_nebismieri polinomia, xolo X(z) _ erTgarovani+−amocanis F ( t)= a(t)F ( t)e.w. kanonikuri amonaxsni, romelic igebacxadi saxiT (kvadraturebSi).amonaxsnis agebisaTvis, romelsac gaaCnia gansazRvrulirigi usasrulobaSi, moiTxoveba davadoT SezRudvebi P(z)polinoms, da aseve b(t) funqcias (ix. n. musxeliSvili, 1966).drekadobis brtyeli Teoriis amocanebis dayvana wrfiviSeuRlebis amocanebamde warmoadgens aseTi amocanebis amoxsnis(gansakuTrebiT Sereuli amocanebis) erT-erT efeqtur meTods.sailustraciod moviyvanoT ZiriTadi Sereuli amocanisamoxsna naxevarsibrtyisaTvis misi wrfiv SeuRlebis amocanamdedayvanis gziT (n.nusxeliSvili, 1966).vTqvaT izotropul sxeuls ukavia qvedanaxevarsibrtyey < 0 , romelsac aRvniSnavT S − -iT. zeda naxevarsibrtye aRvniSnoT+S -iT, namdvili RerZi _ L-iT, L-ze dadebiT mimarTulebad miviRoT_∞-dan +∞-mde.amovideT Zabvebisa da gadaadgilebebis zogadi kompleqsuriwarmodgenis formulebidan, kerZod visargebloT formulebiTYy− iX = Φ( z)+ Φ(z)+ zΦ′( z)+ Ψ(z), (102)y⎛ ∂u∂u⎞2μ ⎜ + i ⎟ = χΦ(z)− Φ(z)− zΦ′( z)− Ψ(z), (103)⎝ ∂x∂x⎠sadac Φ(z) da Ψ(z)_ S − areSi saZiebeli holomorfuli funqciebia,romlebsac didi z -ebisas aqvT saxeX + iY ⎛ 1 ⎞( z)= − + 0⎜⎟ ,2πz⎝ z ⎠X − iY ⎛ 1 ⎞( z)= + 0⎜⎟ ,2πz⎝ z ⎠Φ2Ψ2(X, Y) _ gare Zalvebis mTavari veqtoria, romlebic modebulia L-ze.−S areSi, ori holomorfuli Φ(z) da ψ(ζ) funqciisnacvlad SemoviRoT uban-uban _ holomorfuli erTi Φ(z) funqcia,99

omelic gansazRvrulia rogorcS − -Si, ise S + -Si, amasTan zedanaxevarsibrtyeSi+S , is iqneba gansazRvruli ise, rom misimniSvnelobebi gaagrZeleben analizurad Φ(z)-is mniSvnelobebsqveda naxevarsibrtyeSi−S dautvirTavi ubnebis gavliT (TukiaseTebi arseboben). ganvsazRvroT Φ(z) S + -Si Semdegi formuliTΦ ( z)= −Φ(z)− zΦ′( z)− Ψ(z).es formula iZleva ψ(z) funqciisaTvis gamosaxulebas Φ(z)funqciiT, romelic vrceldeba S + -zec.Ψ ( z)= −Φ(z)− Φ(z)− zΦ′( z);maSasadame, Zabvebis komponentebi gamoisaxeba erTi Φ(z) funqciiT,romelic gansazRvrulia rogorc S + -Si, asevekerZod, gveqneba formulaxolo (103) formulidan miviRebTS − -Sic.Yy − iXy= Φ( z)− Φ(z)+ ( z − z)Φ′ ( z), (104)vTqvaT⎛ ∂u∂v⎞2μ ⎜ + i ⎟ = Φ(z)+ Φ(z)− ( z − z)Φ′ ( z).(105)⎝ ∂x∂x⎠warmoadgens namdvili RerZisL′=n∑k=1a kb ka kbkmonakveTebis erTobliobas, davTqvaT L′-ze mocemulia gadaadgilebaTa komponentebi, xolodanarCen nawilSi L ′ = L − L′_gare Zalvebi. zogadobis daurRvevladSeiZleba CavTvaloT, rom L″-ze mocemuli gare Zalvebi tolianulis (zogadi SemTxveva advilad daiyvaneba amaze).CavTvaloT, rom funqcia Φ(z) uwyvetad gangrZobadiamarcxniv da marjvniv α-ze, garda, SesaZloa, a k, b kwertilebisa,xolo am wertilebis midamoebSiCavTvaloT aseve, romA( z)≤ ,αz −AΦ ( z)≤ ( α

lim( z − z)Φ′ ( z)= 0 .roca z miiswrafis namdvili RerZis t wertilisaken, romelicgansxvavdeba a kda b kwertilebisagan.am daSvebebisas (104) formulidan gamomdinareobs+−Φ ( t)= Φ ( t)L″-zeanu Φ(z) funqcia holomorfulia mTel sibrtyeze, gaWrilia L-isgaswvriv da qreba usasrulobaSi.(105) formulebidan ki gveqneba+−Φ ( t ) = −Φ ( t)+ 2 g′( t)μ ( t L′)∈ (106)sadac g(t) _ mocemuli funqciaa, g ( t)= u(t)+ iv(t), u(t)-Ti da v(t)-TiaRniSnulia L′-ze cnobili, gadaadgilebebis komponentebissasazRvro mniSvnelobebi, CavTvaloT, rom g′ (t)warmoebuliakmayofilebs helderis pirobas.Tu gamoviyenebT (101) formulas, SeuRlebis (106) amocanisamoxsnisaTvis miviRebTμX( z)g′( t)dtΦ ( z ) = ∫ + X ( z)P(z),+πiL X ( t)(t − z)sadac X(z) (kanonikuri amonaxsni) aqvs saxen11− + β − −iβ22⎛ ⎞( z)= ∏(z − ak) ( z − bk) ⎜ β = ⎟k=12 π ⎠ln xX.⎝radganac Φ(z) qreba usasrulobaSi, xolo X(z)-is rigiusasrulobaSi tolia _ n, P(z) polinomis xarisxi ar undaaWarbebdes ( n −1)-s:n−1n−2P( z)= C0z+ C1z+ L + Cn−1.gansaxilveli amocanis wrfivi SeuRlebis amocanazedayvanisas Cven gavawarmoeT sasazRvro pirobebi a kbkmonakveTebisgaswvriv; maSasadame, jerjerobiT Cven movaxerxeTdagvekmayofilebina sasazRvro pirobebiakbk-s gaswvriv mudmivi c kSesakrebebis sizustiT. gvrCeba gaviTvaliswinoT pirobebic c = L = c 0, Tumca, rogorc advili dasanaxia, sakmarisia1=2n=davakmayofiloT pirobebi101

es pirobebi daiyvaneba Semdegze:c =ak∫ + 1( ′ + iv′) dt = g(ak+1)− g(b kbk1= c2= L c n.u ) ( k = 1,2, L , n −1)(107)Tu (107) tolobaSi CavsvamTY u ( t)+ ib(t)-s nacvlad misgamosaxulebas Φ funqciiT, miviRebT n −1wrfiv gantolebaTasistemas.P(z) polinomis C0koeficienti ganisazRvreba gare Zalvebismocemuli (X, Y) veqtoriT:X + iYC0= − .2πTu CavsvamT C0-is mniSvnelobas (107) sistemaSi, miviRebTC L koeficientebis mimarT wrfiv gantolebaTa n −11, C2,, C n −1sistemas. es sistema calsaxad amoxsnadia ZiriTadi Sereuliamocanis amonaxsnis erTaderTobis Teoremis safuZvelze.kompleqsuri cvladis funqciis Teoriis meTodebi,romelTa Sesaxeb zemoT gvqonda laparaki drekadobis Teoriisbrtyel amocanasTan dakavSirebiT, arsebiTad iyo ganviTarebuli i.vekuas gamokvlevebSi kerZo warmoebulebSi diferencialurigantolebebis Teoriis ufro zogadi amocanebisadmi miyenebiskuTxiT. i. vekuas monografiaSi (1948) am TvalsazrisiTgamokvleulia vrceli klasi elifsuri gantolebebisa oridamoukidebeli cvladis SemTxvevaSi da mocemulia avtoris mierganviTarebuli aparatis gamoyenebebi (drekadi cilindrisstacionaruli rxevebi, Txeli firfitebis Runva da sxva).aqve gvmarTebs movixsenioT imave meTodebismravalricxovani gamoyenebebi drekadi garsebis TeoriaSi (i.vekua, a. goldenveizeri, g. savini).kompleqsuri cvladis funqciis Teoriis meTodebTan erTad,romlebic saSualebas iZleva amovxsnaT brtyeli amocanaSedarebiT zogadi saxis areebisaTvis, SeiZleba vipovoT efeqturiamonaxsnebi zogierTi konkretuli formis areebisaTvis kerZo102

xerxebiT, magaliTad furie da melinis integralurigardaqmnebis saSualebiT.furies gardaqmnebi warmoadgens erTob moxerxebul aparatsusasrulo zolis drekadi wonasworobis sxvadasxvagvariamocanebis ganxilvisaTvis. amgvari saxis umartivesi amonaxsnebinapovni iyo jer kidev l.n.j. failonis mier. es meTodika,romelmac miiRo farTo ganviTareba sabWoTa mecnierebisnaSromebSi, ocdaaTiani wlebis bolos iyo ganzogadebuli daSejamebuli p.papkoviCis cnobil monografiebSi (1939, 1941).SemdgomSi sxvadasxva avtorebis mier ganxiluli iyomniSvnelovani raodenoba axali amocanebisa, romlebic Seexebodazolis, naxevarzolis deformaciebs, romlebic Seesabamebodnenfenovan garemos da anizotropul tanebs, Tbur Zabvebs da sxva.dainteresebul pirebs SeuZliaT dawvrilebiT gaecnon d.Sermanis (1962) mimoxilviT naSromebs. g. popovisa da n.rostovcevisa (1966), s.lexnickis (1957) da m. Seremetievis (1968)monografiebs.aqve mivuTiTebT i. alperinis (1930), m. belenkovis (1952) das.birmanis (1954) statiebs, romlebic Seexeba Sereul amocanebsusasrulo zolisaTvis, da aseve i. markuzonisa (1963), v.tonoianis naSromebs, romlebSic Sereuli amocanebis zogierTiklasebi naxevarsibrtyisa, zolisa da kvadrantisaTvisgadawyvetilia wyvili an sammagi tolobebiT, romlebicdakavSirebulia furies gardaqmnebTan.rigi saintereso amocanebi amoixsna bipolarulkoordinatebSi, furies integralebis saSualebiT. msgavsi saxisamocanebi, romlebic ZiriTadad wriul `namgalasTan~ arisdakavSirebuli, ganixileboda i. ufliandis (1950, 1963), g. savinis(1951), m. savrukis (1957), v. eganianis (1959, 1964) da sxva avtorebismier.drekadobis Teoriis zogierTi brtyeli amocana usasrulosolisaTvis zust gadawyvetas Rebulobs melinis integralurigardaqmnebis saSualebiT. Tavdapirveli gamokvlevebi am wris103

sakiTxebisa ekuTvniT i. bratcsa da v.abramovis (1937). solzeSeyursuli Zalis moqmedebis Sesaxeb amocana pirvelad ganixilesa. luriem da b. braCkovskim (1941). anizotropuli solis amocanagamokvleeuli iqna p. kuvarevis (1941) mier. miTiTebul amocanebzebibliografia moyvanilia i. ufliandis (1963) wignSi.furiesa da melinis integralur gardaqmnaTa meTodisganviTareba koSis tipis integralebis aparatTan SeTavsebaSiSedis s. belonosovis (1962) naSromebSi, romlebic ganekuTvnebianareebs kuTxis wertilebiT da kerZod zolsa da sols.2.2-is daskvnebirogorc zemoT vaCveneT, musxeliSvilis gamokvlevebiproblemebis farTo jgufs moicavs. aRniSnulma Sromebma didigavlena moaxdines meqanikisa da maTematikis rigi dargebisSemdgom ganviTarebaze.n. musxeliSvilis meTodebma drekadobis brtyel TeoriaSigamoyeneba da Semdgomi ganviTareba hpoves s.mixlinis, d.Sermanis,g.savinis, d.vainbergisa da sxva naSromebSi am meTodebissaSualebiT amoxsnilia mravali amocana, romlebmac farTogamoyeneba hpoves teqnikaSi. l.galinis, a.kalandias, i.qarcivaZis,i.Staermanis, r.bancuris da sxva naSromebSi n.musxeliSvilisSedegebma Semdgomi gamoyeneba da ganviTareba hpova sakontaqtoamocanebis TeoriaSi.gamokvlevebi Zelebis grexisa da Runvis amocanebSisxvadasxva mimarTulebiT gagrZelebul iqna a. gorgiZis, a. ruxaZisda sxvaTa mier.104

2.3. myari deformadi sxeulis meqanikis ganviTarebisistoria Txelkedlian sivrciT sistemebisaTvis2.3.1. brtyeli drekadobis Teoriis amocanebisgamokvlevebis ZiriTadi SedegebinaSromebi, romlebsac Cven SevexebiT, ZiriTadad mWidrodaadakavSirebuli kompleqsuri cvladis meTodebTan da amTvalsazrisiT isini gamodgeba maTi gamoyenebis sailustraciod daSemdgomi ganviTarebisaTvis.ZiriTadi amocanebis amoxsna erTgvarovani garemosaTvis.konkretuli Sinaarsis pirveli Sedegebi, romlebic ganekuTvnebabrtyeli profilebis wonasworobas miRebuli iyo g. kolosovisada n. musxeliSvilis mier.92-Si miTiTebuli meTodiT n. musxeliSvilma mogvcapirveli da meore ZiriTadi amocanebis martivi amonaxsni wris,wriuli rgolisa da usasrulo sibrtyisTvis wriuli naxvretiT.damuSavebuli iqna mravali kerZo magaliTi sxvadasxva saxis garezemoqmedebebisaTvis. amgvari saxis areebisaTvis ar moiTxovebawinaswari konformuli asaxva. gamoiyena ra konformuli asaxva,musxeliSvilma gadawyvita im droisaTvis rTuli amocana mTlianielifsis wonasworobis Sesaxeb. mogvianebiT igive amocanas igiwyvetda sxva xerxebiT (ix. 96).xarisxovani mwkrivebis meTodiT iyo gamokvleuli amocanaTanafokusuri elifsuri rgolis Sesaxeb (a. kalandia, 1953). amamocanis efeqturi amoxsnis algoriTmi kidev ufro adremiTiTebuli iqna m. Seremetievis mier, romelic iyenebdafunqcionaluri gantolebebis meTods konformul asaxvebTannaerTSi (ix. 93).dasaxelebuli meTodi yvelaze moxerxebuli aRmoCndacaladbmuli areebisaTvis. rogorc es zemoT iyo aRniSnuli, masyovelTvis mivyavarT efeqtur amoxsnamde, Tu ares asaxvaganxorcielebulia racionaluri funqciiT. meTodis pirveligamoyenebebi iyo miTiTebuli TviT n.musxeliSvilis mier,105

omelmac mogvca Caketili amonaxsnebi ZiriTadi amocanebis rigikonkretuli SemTxvevebisaTvis. amocanaTa am seriidan CvengamovyofdiT wriuli diskis wonasworobas konturuliSeyursuli datvirTvebis moqmedebis qveS da usasrulo firfitaselifsuri naxvretiT. musxeliSvilis Sedegebi, romelzedac aqiTqva, miRebuli iyo ociani da ocdaaTiani wlebis SromebSi (maTSoris gansakuTrebiT unda aRvniSnoT misi memuari gamocemuli1922 wels). yvela es Sedegebi sxvebTan erTad, romelic ekuTvnisimave avtors, dawvrilebiTaa gadmocemuli, ara erTxel zemoTcitirebul, n. musxeliSvilis monografiaSi.aRvniSnoT amave meTodis kidev erTi mniSvnelovani gamoyeneba,romelic miuTiTa g. savinma. ganvixiloT amocana Zabvebiskoncentraciis Sesaxeb usasrulo firfitaSi, romelicdasustebulia raime naxvretiT. miviCnevT ra naxvretis konturssworxazovan mravalwaxnagad, avsaxoT wris Siga nawili naxvretisgareTa areze Svarci-kristofelis integralis saSualebiT.gavSliT ra am integrals ζ-s xarisxebis mwkrivad da davtovebTra mwkrivSi misi pirveli wevrebis sasrulo ricxvs, CvenmiviRebT miaxloebiT amonaxsns, romelsac gadayavs wrexazisawyisi konturis maxlobel mrudSi⎛ 1nk ⎞z = ω ( ζ ) = C⎜+ ∑C kζ ⎟(108)⎝ ζ k=1 ⎠saxiT, an kerZo SemTxvevaSisadac C ,⎛z = C⎜⎝1ζn ⎞+ mζ⎟ , (109)⎠C k, m − raime mudmivebia. Tu SevcvliT (108)-Si C, C k , nmudmivebs SegveZleba miviRoT naxvretebi wris, elifsis, ovalis,mrudwira samkuTxedisa da oTxkuTxedis da a.S. formiT. (108)asaxvisas meTods dauyovneblad mivyavarT amonaxsnamde CaketilisaxiT, es ki iZleva miTiTebuli saxis amocanebis miaxloebiTiamoxsnis SesaZleblobas.am gziT g. savinma da misma mowafeebma ganixiles bevrikonkretuli amocana erTgvarovan velSi, sxvadasxvagvari formisa106

da konfiguraciis naxvretebis dros Zabvebis koncentraciisSesaxeb. am amocanebis amonaxsnebi miyvanilia ricxviTSedegebamde, romlebic warmodgenilia cxrilebisa da diagramebissaxiT. garda amisa, gansakuTrebulad mniSvnelovan SemTxvevebSi,agebulia konturuli Zabvebis grafikebi. msgavsi saxiT ixsnebasavinis mier amocanebi, naxvretis mqone Txeli firfitebis RunvisSesaxeb, romlebic moqceulia usasrulobaSi momentebisa danormaluri Zalvebis moqmedebis qveS. es Sedegebi dawvrilebiTgadmocemulia g.savinis wignSi (1951), romelmac SeasrulamniSvnelovani roli am wris sakiTxebis Semdgom damuSavebaSi.Zabvebis Sesaxeb sakiTxebiT naxvretebis mqone firfitebSi,roca maT mrudwiruli mravalkuTxedis forma aqvT, g. savinTanerTdroulad dakavebuli iyo m. neimani (1937, 1958), romelmacgamoiyena originaluri midgoma miaxloebiTi anasaxebis arCevisas;is ZiriTadad Seiswavlida Zelebis grexas, romlebicdasustebulia amonaCarxebiT.93-Si gadmocemuli meTodi SeiZleba gamoyenebuli iqnasraRac saxecvlilebebiT naxevradusasrulo areebis SemTxvevisas,roca garemos sazRvrad gvevlineba mrudi, romelic gvcildebausasrulobaSi orive mxares. am SemTxvevaSi ufro mosaxerxebeliavisargebloT asaxviT naxevarsibrtyeze meTodis gamoyeneba zogaddasmaSi naCvenebia n. musxeliSvilis (1966) monografiaSi, sadacmoyvanilia agreTve amgvari saxis zogierTi kerZo amocanisamoxsna.gamoyenebisaTvis gansakuTrebul interess warmoadgensnaxevarsibrtyeebi, romelic dasustebulia amonaWeriT an aqvsSverilebi sworxazovan sazRvarTan. aseTi tipis amocanebsukanasknel xanebSi didi yuradReba eTmoba.n. kurdinis midgoma am amocanebisadmi warmogvidgebayvelaze ufro warmatebulad. iyenebda ra musxeliSvilis meTods,man moaxerxa detalurad gaerCia miTiTebuli saxis zogierTisaintereso SemTxveva (1962).funqciis Teoriis meTodebis gamoyenebis SesaZlebloba107

firfitebis Runvis amocanebSi pirvelad ilustrirebuli iqna a.luries (1928) naSromSi, sadac ganixileboda firfitadayrdnobili kideebiT, romlis Sua zedapiris are konformuladaisaxeboda wreze racionaluri funqciis meSveobiT. ufrodawvrilebiTi Seswavla am sakiTxisa iyo Catarebuli mogvianebiTa. kalandias (1953) mier. a. luries (1940) sxva naSromSi mocemuliaigive meTodiT miRebuli amonaxsnebi Runvis Teoriis samiZiriTadi amocanis wris SemTxvevisaTvis. aq, iseve rogorc imaveavtoris wina naSromSi, gamoiyeneboda musxeliSvilis meTodi (93).zemoT citirebuli s. lexnickis naSromi (1938) Seicavskompleqsuri cvladis meTodebis gamoyenebas firfitebis RunvisamocanebSi. masSi gamoiyvaneba ZiriTadi sidideebis zogadikompleqsuri warmodgenebi izotropuli da anizotropuliSemTxvevebisaTvis, saboloo saxiT formulirdeba ZiriTadiamocanebi kompleqsuri cvladis terminebSi da moicema maTiamonaxsnebi zogierT kerZo SemTxvevebSi.a. luriesa da s. lexnickis miTiTebuli naSromebiT daiwyofirfitebis Runvis TeoriaSi intensiuri gamokvlevebi.93-is meTodiT m. fridmanma (1945) ipova firfitis RunviszogierTi konkretuli amocanis amonaxsnebi, romelTac aqvTmrudwiruli naxvreti, iRuneba momentebiTa da ZalvebiT, romlebicmis kidezea modebulia.gansakuTrebuli yuradReba eTmoboda kideebiT dayrdnobilifirfitis wonasworobis amocanas. mis Seswavlas mieZRvnaz.halilovis (1950), m. fridmanis (1952), d. Sermanis (1959),a.kalandias (1953) naSromebi.funqciis wrfivi SeuRlebis meTodi (ix. 97) aRmoCnda erTobmoxerxebuli saSualeba amocanebis zogadi gamokvlevebisaTvis, daaseve maTi efeqturi amoxsnisaTvis specialur SemTxvevebSi. masgaaCnia aSkara upiratesoba sxvebTan SedarebiT Sereuli dasakontaqto amocanebis Seswavlisas, sadac mniSvnelovaniaSegveZlos amoxsnis TaviseburebaTa gamoyofa. am tipis amocanebsganvixilavT qvemoT.108

am meTodis gamoyeneba brtyeli amocanebisadmi pirveladmiTiTebuli iyo n. musxeliSvilis naSromSi (1941), sadacganixileboda drekadi naxevarsibrtyis SemTxveva. ZiriTadiamocanebis amonaxsnebi am SemTxvevaSi napovni iqna martiv dafriad efeqtur formaSi. am meTodis Semdgomi arsebiTiganzogadoeba SemoTavazebuli iyo i. qarcivaZis (1943) mier,romelmac is ganavrco wriuli ares SemTxvevaze, da agreTvewreze asaxvis ufro zogad SemTxvevaze racionaluri funqciissaSualebiT. imave avtors ekuTvnis miTiTebuli areebisaTvismeTodis gamoyeneba konkretuli amocanebis amosaxsnelad.qarcivaZis Sedegebi dawvrilebiTaa gadmocemuli n.musxeliSvilis wignSi (1966). wriuli xvrelis mqonesibrtyisaTvis Sereuli amocana ganixileboda b. mincbergis (1948)mier.amave meTodiT n. musxeliSvilma miuTiTa brtyelidrekadobis Teoriis mesame ZiriTadi amocanis amoxsnaze CaketilisaxiT.xisti profiliT SeuRlebis amocana sxva meTodebiTSeiswavla g.poloJim. amocanis sasazRvro pirobebi winaswar iqnamoqceuli zogierTi gardaqmnebis qveS, romlebmac gaamartives ampirobebis forma sazRvris sworxazovan ubnebze. swored amanmisca avtors SesaZlebloba mieTiTebina amocanis cxadi amoxsna,jer amozneqili mravalkuTxedebisaTvis (1949, 1950), xolo Semdeg,erTob daxvewili gamokvlevebis Semdgom kuTxis wertilebSiZabvebis qcevis Sesaxeb, gadaadgilebebis veqtoris uwyvetobispirobisas, yvelaze zogadi saxis mravalkuTxedebisaTvis, da asevenebismieri mravalkuTxa naxvretis mqone usasrulonaxevarsibrtyisaTvis (1957).Seiswavlida ra ZiriTad brtyel amocanebs kuTxeebis mqonecaladbmuli areebisaTvis s. belonosovma (1954, 1962) SemogvTavazamaTi iseTi amoxsnis meTodi, romelic saSualebas iZleva miecesTeoriuli dasabuTeba kuTxeebis momrgvalebaze dafuZnebul,miaxloebiTi amoxsnis praqtikul xerxs: mocemuli ares109

naxevarsibryeze Re ζ > 0 konformuli asaxva saSualebas iZleva ϕda ψ kompleqsuri potencialebis mosaZebnad gamoviyenoTlaplasis calmxrivi gardaqmnis aparati. miRebuli meTodis,analogiurad n.musxeliSvilis mier, igeba sakmaod martivistruqturis integraluri gantolebebi, romlebic moixmarebagarkveuli azriT areebisaTvis, kuTxis wertilebiT. Tu L konturiar Seicavs kuTxis wertilebs da saerTod sakmarisad gluvia,maSin integraluri gantolebis guli aris fredholmuri, xolouban-uban gluvi konturis zogad SemTxvevaSi is miekuTvnebakarlemanis gulebis tips.s. belonosovis integraluri gantolebebi, rogorc manaCvena (1962), amoxsnadia ZiriTad amocanaTagan TiToeulisaTvis.usasrulo solis an zolis kerZo SemTxvevebSi integralurigantolebebi ixsneba kvadraturebSi, rasac mivyavarT amSemTxvevaSi sasrulo saxiT amocanebis amoxsnamde. s.belonosoviscitirebul wignSi gansazRvrulia agreTve areebis klasi,romlebisTvisac ZiriTadi amocanebi amoxsnadia kvadraturebSimiTiTebuli meTodiT. es klasi, garda areebisa, romlebic formiTmaxlobelia solTan, zolTan da hiperbolasTan, Seicavs TavisTavSi agreTve wriul koncentrul rgols.zemoT gadmocemuli d. Sermanis meTodi (95) mis mierTavdapirvelad SemoTavazebuli iyo oradbmuli profilebisgansazRvruli klasis, grexisa da Runvis amocanebis amoxsnisaTvis.brtyeli deformaciisadmi SefardebiT is iyo ilustrirebuli(1951) naxevarsibrtyis magaliTad, romelic dasustebuliaaraerTnairi ori wriuli naxvretiT. Sermanis ufro gviandelgamokvlevebSi mimdinareobda meTodis arsebiTi gardaqmnebi,ramac moxsna didi moculobis Sualeduri gamoTvliTioperaciebis Catarebis aucilebloba. Sedegad amoxsnis procesigaxda Tvalmisawvdomi da miiRo Tavis ZiriTad nawilSirekurentuli Tanafardobebis xasiaTi.d. Sermanisa da misi mowafeebis mravalricxovan naSromebSi,romlebic gamoqveynebulia ukanasknel wlebSi, mocemulia meTodis110

gamoyeneba brtyeli deformaciis Sesaxeb konkretul amocanebSi.ganxiluli iyo amocanebi wonadi naxevarsibrtyis Sesaxeb orinaxvretiT (wriuliT da elifsuriT), romlebic ganlagebuliagaremos wrfivi sazRvridan mniSvnelovani daSorebiT; drekadiwris Sesaxeb naxvretiT, sakmaod zogadi moxazulobisa;naxevarsibrtyis Sesaxeb naxvretiT, romlis kidis gaswvrivCarCilulia rgoli sxva masalisagan, da sxva analogiuriamocanebi. Sedegebis safuZvliani mimoxilva integralurigantolebebis meTodis gamoyenebaze sruli bibliografiiTmoyvanilia d. Sermanis narkvevSi (1962).mravalbmuli ares calkeuli SemTxvevisaTvis gamoiyenebodaSvarcis ganzogadebuli algoriTmi, romelic zogad formaSiganaviTara s. mixlinma (1949) ZiriTadi biharmoniuli amocanismimarT. meTodis pirveli ilustracia gakeTda imave avtoris mier(1934) elifsuri naxvretis mqone, wonadi naxevarsibrtyismagaliTze, roca usasrulobaze Zabvebi ganawilebuliahidrostatikuri kanoniT.Svarcis mixedviT TandaTanobiT miaxloebebis krebadobagamoikvleoda ares mimarT zogierTi SezRudvebisas, s. mixlinisada a. gorgiZis naSromebSi. meTodis krebadoba zogad SemTxvevaSidadgenili iqna s.sobolevis (1936) mier.Svarcis algoriTms ar gaaCnia swrafi krebadoba, romelicunda gvaxsovdes meTodis praqtikuli gamoyenebisas. miuxedavadamisa rig SemTxvevebSi man SeiZleba mogvces damakmayofilebeliSedegebi. magaliTebad gamogvadgeba a. kosmodamianskis naSromebi(1961, 1964), romlebic exeba ori araerTnairi naxvretis SemTxvevasusasrulo garemoSi.firfitaSi, bevri naxvretebiT, Zabvebis Seswavlisas erTerTZiriTad sakiTxs warmoadgens mocemuli naxvretis siaxlovesgaremos Sesustebis, romlebic gamowveulia mezobeli naxvretebisarsebobiT, xarisxis gansazRvra. am sakiTxebs, romlebic iwvevendid interess samTo saqmeSi, mieZRvna d. Sermanisa da misi111

mimdevrebis naSromebi, romelTa Sesaxeb zemoT iTqva. mivuTiToTzogierT ganzogadebaze am mimarTulebiT.im SemTxvevaSi, roca garemo Sesustebulia nebismierisasrulo ricxvis naxvretiT a. kosmodamianski (1961, 1962) iyenebdabubnov-galiorkinis meTods. saZiebeli ϕ da ψ kompleqsuripotencialebisaTvis is sargeblobda ganusazRvreli koeficientebismqone, specialuri saxis funqciebis usasrulo jamis saxiTwarmodgenebiT da iRebda miaxloebiTi amoxsnisaTvis algebrulgantolebaTa sasrulo ricxvis sistemebs, meTods mivyavarTgansakuTrebiT karg Sedegebamde wriuli naxvretebis SemTxvevaSi.miaxloebis rigis usazRvro gazrdisas algebrulisistemebi xdeba usasrulo. imave avtorebis gamokvlevebmagviCvena, rom am sistemebs gaaCniaT xelsayreli Tvisebebinaxvretebis erTmaneTTan, rogorc gnebavT, mcire siaxlovisasacki. arawriuli mrudwiruli naxvretebis SemTxvevaSimizanSewonili xdeba im meTodebis gamoyeneba, romlebic axlosaaideaSi n. musxeliSvilis meTodTan (a. kosmodamianski, 1962).miTiTebuli miaxloebiTi xerxebi kosmodamianskis da zogierTisxva avtoris mier iyo gamoyenebuli amocanebis amoxsnisaTvisrig konkretul SemTxvevebSi.g. muxarinovma (1937, 1939), gamoiyena ra mimdevrobiTimiaxloebis raRac algoriTmis analogi, damuSavebuli g.goluzinis mier dirixles amocanisaTvis, Seiswavla amocanafirfitisa da diskisaTvis, roca garemo Sesustebulia wriuliformis, nebismierad ganlagebuli, nebismieri sasrulo ricxvisnaxvretebiT.mniSvnelovan interess iwvevs drekadobis Teoriis perioduliamocana. warmovidginoT SemousazRvreli erTgvarovani garemo,romelic Sesustebulia erTnairi da periodulad ganlagebulinaxvretebis usasrulo rigiT. davuSvaT, rom yvela es naxvretimoeqca erTnairi gare Zalvebis moqmedebis qveS da maTi centrebiganlagebulia erTsa da imave sworze. naxevarsibrtyis SemTxvevaSiiTvleba, rom centrebis swori paraleluria naxevarsibrtyis112

sazRvrisa da imyofeba misgan manZilze, romelic mniSvnelovnadaRemateba naxvretis zomebs.geometriuli da Zalovani simetriis erToblivi faqtorebisarsebobas mosdevs gadaadgilebebisa da Zabvebis periodulobacvladis (nivTieri) mimarT, romelic icvleba centrebis xazisgaswvriv. es perioduloba iZleva amocanis reducirebissaSualebas msgavs amocanaze ori funqciis moZebnisa, romlebicholomorfulia romeliRac Sekruli konturis gareTa areSi.mosazrebebi, romlebmac migviyvanes (92) integralurgantolebebamde, aqac miudgeba, rac gvaZlevs SesaZleblobasavagoT amonaxsni fredholmis integraluri gantolebebisaTvis,romlebic yovelTvis wydeba erTaderTi saxiT. es gaakeTa g.savinma (1939) (ix. agreTve s. mixlini, 1949).funqcionaluri gantolebebisa da xarisxovani mwkrivebismeTodebis erToblivi gamoyenebiT xerxdeba rig SemTxvevebSiavagoT amocanis efeqturi amonaxsni. mivuTiToT zogierTnaSromze am mimarTulebiT.d. Sermanma (1961) gamoikvlia Zabvebis veli wonad garemoSi,romelic Sesustebulia wriuli da kvadratuli formis,periodulad ganlagebuli naxvretebiT. amocana ixsneboda wrfivalgebrul gantolebaTa usasrulo sistemaze dayvaniT.amonaxsnis raodenobrivma analizma avtors saSualeba miscakvaldakval mihyoloda naxvretebis axlos Zabvebis ganawilebas εricxobrivi parametris, romelic axasiaTebda ares fardobiTzomebs cvlilebis mniSvnelovan diapazonSi; ar warmoadgensgamonakliss erTmaneTTan axlos mdebare naxvretebis SemTxvevac.zogadi formis mrudwiruli naxvretebiT perioduliamocana ganixileboda kidev ufro adre i. voroviCisa da a.kosmodamianskis naSromebSi (1959). saZiebeli kompleqsuripotencialebisaTvis avtorebis mier SemoTavazebuli iyozogierTi integraluri warmodgenebi, romlebic gamosaxavdnenmaT xsva analizuri funqciebiT, romlebic holomorfulebiasibrtyeSi erTi naxvretiT. SemdgomSi am ukanasknelTa113

sapovnelad gamoiyeneboda mcire parametris meTodi, da amocanadadioda erTi tipis amocanaTa Tanmimdevrobaze caladbmuliaresaTvis. meTodis krebadoba ar gamokvleula. ricxviTigamoTvlebis dawvrilebiTi analizi Catarda elifsuri naxvretebisSemTxvevaSi, roca firfita iWimeba usasrulobaze ZalvebiT,romlebic mimarTulia centrebis xazisadmi nebismieri kuTxiT. ammidgomis zogierTi Semdgomi ganzogadeba mocemulia a.kosmodamianskis (1965) mier.unda aRiniSnos, rom drekadobis Teoriis brtyeliperioduli amocanapirvelad ganxiluli iqna v. naTanzonis (1935) mier, romelicikvlevda usasrulo sxeulSi ornairad perioduli wriulinaxvretebis sistemis SemTxvevas.ufro dawvrilebiT cnobebs perioduli amocanis Sesaxebdainteresebuli piri ipovis zemoT citirebul d. SermanismimoxilvaSi (1962).ukanasknel wlebSi didi yuradReba eqceoda brtyeli amocanebisamoxsnis efeqturi meTodebis Ziebas, roca drekadobis ZiriTadikanoni arawrfivia, magram deformaciis simciris Sesaxeb daSvebaSenarCunebulia. gansakuTrebul interess iwvevdnen sakiTxebi,romlebic dakavSirebulia naxvretebis mqone firfitebsa dagarsebSi Zabvebis koncentraciis gansazRvrasTan.Tu drekadobis kanonis arawrfivobas davaxasiaTebT mcirericxviTi parametriT, maSin biharmoniuli gantolebis nacvlad amSemTxvevaSi ZabvaTa funqciisaTvis miviRebT meoTxe rigisarawrfiv diferencialur gantolebas kerZo warmoebulebSimTavari biharmoniuli wevriT. am gantolebis integrebisaTvisSesabamisi sasazRvro pirobebisas gamoiyeneboda mcireparametris meTodi, amasTan drekadobis kanonis gadaxrawrfivisagan da naxvretis gadaxra wriulisagan miCneuli iyoerTdroulad mcireebad. Tu gavSlidiT ZabvaTa funqcias,gadaadgilebaTa veqtoris koordinatebs, da agreTve funqciebs,romlebic figurireben amocanis sasazRvro pirobebSi mwkrivebad114

ori parametriT, romlebic zemoT dasaxelebul gadaxrasaxasiaTeben, miviRebT Tanamimdevrobas zogierTi biharmoniuliamocanebis wriuli naxvretis mqone sibrtyisaTvis.am gziT iqna amoxsnili arawrfivi drekadobis Teoriisrigi konkretuli amocanebisa.ricxviTi gaangariSebebi gviCveneben, rom fizikuriarawrfivobis gaTvaliswinebas mivyavarT ufro Tanabar, wrfivTeoriasTan SedarebiT, Zabvebis ganawilebasTan naxvretebTanaxlos; ZabvaTa koncentraciis koeficienti ufro mcire xdeba.am mimarTulebis Sedegebis ufro dawvrilebiT gacnobaSeiZleba g.savinis (1965), a. gunias, g. savinis da i. cumpalas (1964)naSromebSi.2.3.2. uban-uban erTgvarovani garemo. Semagrebuli dagaZlierebuli firfitebiuban-uban erTgvarovans Cven vuwodebT drekad garemos,romelic Sedgenilia erTgvarovani nawilebis sasruloricxvisagan, romlebic sxvadasxvagvaria formiTa da drekadiTvisebebiT. sxvadasxva nawilebis gaerTianeba SeiZleba iyosrogorc bunebrivi, aseve xelovnuric. ukanasknelni yovelTvisemsaxurebian konstruqciis mzidunarianobis gaZlierebis mizansda xSirad gamoiyeneba sainJinro praqtikaSi.warmovidginoT sasrulo an usasrulo firfita, romelsacgaaCnia naxvretebis raRac ricxvi, sadac Cadgmulia mTliani anTavis mxriv naxvretebiT dasustebuli, sayelurebi sxvamasalisagan. sayeluris firfitasTan SeerTebisas is SeiZlebaiyos CarCiluli naxvretSi fargis gaswvriv, Cawnexili, andavTqvaT, Casmuli masSi cxel an civ mdgomareobaSi, yovelTvis, Tusayeluri ar aris CarCiluli, ivaraudeba, rom erTmaneTTanSeuRlebuli drekadi nawilebis konturebi modis TanxvedraSiRreCoebis gareSe da Sekavebulia erTi meores srialisagan.115

am TavSi damatebiT miviCnevT, rom sxeulebisurTierTSexebuli zedapirebi arsad ar cildebian erTmaneTsdeformaciis gamo.Semagreba SeiZleba, ra Tqma unda, ara marto naxvretiskideebis. firfitebi SeiZleba gavamagroT rgolebiT mis nebismierkideze, da aseve Siga nawilebSi, romlebic ar emijnebiansazRvars. ukanasknel SemTxvevaSi laparakoben firfitisgamagrebaze sixistis wiboebiT.Sedgenili sxeulis mTliani L sazRvari iqneba Sedgenilifirfitis gare konturisagan (Tu rasakvirvelia is ar vrceldebausasrulod yvela mxares), Seumagrebeli naxvretebiskonturebisagan da bolos Casadgmeli sayelurebis Sidakonturebisagan, Tu aseTebi arseboben. sxeuli SeiZlebaganicdides nebismier zemoqmedebas rogorc SigniT, asevesazRvarzec.firfitis Seumagrebel kideze sasazRvro pirobebi iqneba,aSkaraa, Cveulebrivi, Sesabamisi masze mocemuli Zalovanizemoqmedebis reJimisa an misi Camagrebis xasiaTisa. sxvadasxvagaremoTa xazze pirobebi ki iqneba sxvadasxvagvari im nawilebisSeerTebis xerxTan damokidebulebiT, romlebic mas esazRvrebian.magaliTad, SemTxvevaSi, roca firfitaSi yvela naxvreti,romlebic ikaveben sasrulo mravalbmul S ares sazRvriTL L , Sevsebulia mTliani diskebiT sxvadasxva= L1 + L2+ + Lm+1masalisagan, romlebic firfitasTan mirCilulia naxvretebisfarglebis gaswvriv, xolo daZabuli mdgomareoba gamowveuligare ZalebiT, romlebic modebulia mxolod firfitis garekonturze, Cven gveqneba analizuri funqciebis Teoriis Semdegiamocana:( t ) + tϕ′( t)+ ψ ( t)= f ( t)ϕm+ 1ϕ ( t)tϕ′( t)+ ψ ( t)= ϕ ( t)+ tϕ′( t)+ ψ ( t)L -ze, (110)+kk kkx 1ϕ( t)− [ tϕ′( t)+ ψ k( t)]=μ μL -ze, (111)116

[ tϕ′( t)+ ψ ( )]xk1ϕk( t)−k ktμ μ=kkkL -ze ( k = 1,2, L,m)(112)sadac ϕ da ψ holomorfulebia S areSi, xolo ϕkda ψkholomorfulebia sasruloS − kareze, romelic SemosazRvruliaL ( k = 1,2, L,m)konturiT. (110) sasazRvro pirobis azri naTeliakwinamdebaresagan. (111) da (112) tolobebi ki gamoxatavengadaadgilebaTa da ZabvaTa veqtorebis komponentebis uwyvetobisaSkara pirobebs garemoTa gamyof xazze gadasvlisas. k indeqsimiwerili aqvT sayeluris masalis drekad elementebs, romlebsacukaviaTS − kare.erT-erTi adrindeli gamokvleva, romelic exebodaaraerTgvarovani drekadobis sxeulebs da romelicSesrulebulia kompleqsuri cvladis meTodebis bazaze, iyos.mixlinis naSromi (1935), romelSic 94-Si naxsenebi Svarcisgulis saSualebiT, Seiswavleboda zogadi amocana uban-ubanerTgvarovani garemos Sesaxeb, integralur gantolebaTameTodiT. zogierT kerZo SemTxvevebi ganxiluli iyo efeqturisaxiT imave avtoris sxva naSromSi (1934).SemdgomSi amocanebis gamokvlevam araerTgvarovanidrekadobiT daiwyo swrafad ganviTareba. am areSi mniSvnelovaniwarmatebebi iqna miRweuli ukraineli mecnierebis mier, sadacSesabamisi sakiTxebi didi xnis ganmavlobaSi iyo bevri avtoriskvlevis sagani. am gamokvlevaTa Sedegebi gadmocemulia g. savinis(1951), d. vainbergis (1952), m. Seremetievis (1960) da v. savinisa da n.fleiSmanis monografiebSi. qvemoT Cven mokled SevexebiT ZiriTadSedegebs.daviwyoTYSedarebiT martivi SemTxvevidan, roca ZiriTadifirfita da drekadi sayelurebi, romlebic naxvretebSi idgmeva,damzadebulia erTi da imave masalisagan. am SemTxvevaSi Cven undadavuSvaT, rom dauZabav mdgomareobaSi sayelurebis konturebiramdenadme gansxvavdeba Sesabamisi naxvretebis konturebisagan.gamosayeneblad gansakuTrebiT sainteresoa SemTxveva, roca117

sayelurebi Cawnexilia an Casmulia naxvretSi mocemuli drekadiWeqiT.am amocanis sasazRvro pirobebi miiReba (110)-(112)-dan, Tu(112)-is marjvena nawils mivuerTebT mocemul funqcias, romelicgamosaxavs drekadi gadaadgilebebis naxtomis arsebobas dagaviTvaliswinebT, garda amisa, rom sxeulis drekadi Tvisebebiyvelgan erTnairia.am amocanis amoxsnis zogadi xerxi SemoTavazebuli iyo d.Sermanis (1940) mier. es xerxi damyarebulia funqciis analizurgagrZelebaze, 95-Si gadmocemulis msgavsze. am xerxis Tanaxmadgansaxilveli amocana daiyvaneba Cveulebriv brtyel amocanazesruli Sedgenili aresaTvis, gayofis xazze raime pirobebisgareSe. amasTan, Tumca, axlad miRebul amocanas eqneba (garekonturze) ramdenadme Secvlili sasazRvro piroba; tolobismarjvena nawilSi, romelic am pirobas warmoadgens, gaCndebadamatebiTi Sesakrebi, romelic gamosaxavs raime fiqtiurzemoqmedebas mTel sistemaze mTlianobaSi.SemTxvevaSi, roca CanarTebs aqvT wriuli forma, axlaxannaxsenebi Semasworebeli wevri SeiZleba warmodgenili iqnascxadi saxiT. roca gadaadgileba mimarTulia normalze da misisidide mudmivia mas aqvs gansakuTrebiT martivi saxe, romelicpraqtikaSi xSirad gvxvdeba.saboloo angariSSi wriuli CanarTebis SemTxvevaSiamocanis amoxsna aq miiyvaneba bolomde Sedgenili areebisaTvis,romlebic aisaxebian wreze racionaluri funqciis meSveobiT. amgziT iqna ganxiluli konkretuli amocanebis didi ricxvi.dawvrilebiTi miTiTebebi Sesabamis publikaciebze moipovebazemoT citirebul d. Sermanis mimoxilvaSi (1962).sxvagvaradaa saqme sxva drekadi maxasiaTeblebis mqoneCanarTebis SemTxvevaSi. xisti CanarTis ganxilvas. aSkaraa, arSemoaqvs araviTari garTulebebi, vinaidan am SemTxvevaSi Cvensaqme gveqneba Cveulebriv brtyel amocanasTan naxvretiskonturze mocemuli drekadi gadaadgilebebisas (meore ZiriTadi118

amocana). sxvadasxva masalebisagan drekadi CanarTebis Sesaxebamocana gacilebiT rTulia.es amocana erTi CanarTisaTvis, roca k=1, (110)-(112)-Sigamoikvleoda meTodiT, romelicDanalogiuria 3.3.5-Si dasaxulis(d. Sermani, 1958) amjerad firfitis mTelL 1+ L 2− ze sazRvarzeSemosayvani damxmare ω(t) funqciis sapovnelad, avtorma miiRofredholmis integraluri gantoleba da mogvca misi gamokvleva,CanarTis mqone eqscentruli rgolis kerZo SemTxvevaSi, romelicganixileboda sailustraciod, integraluri gantoleba icvleba,rogorc (ix. 95) wrfiv algebrul gantolebaTa usasrulosistemiT, romelic iZleva saSualebas miviyvanoT amoxsna ricxviTSedegebamde.erTmaneTSi Tanmimdevrulad Calagebuli ucxo wriulikoncentruli rgolebis SemTxveva, rogorc adre iyo aRniSnuli,advilad eqvemdebareba ganxilvas xarisxovani mwkrivebismeTodiT.es meTodi funqcionalur gantolebebTan erTobaSisaSualebas iZleva ganvixiloT amocana rgoluri SemagrebebiTramdenamde zogad SemTxvevaSi, magaliTad, roca usasruloerTbmuli are, romelic ukaviaT SeuRlebul sxeulebs, aisaxebawris garegnobaze racionaluri funqciis saSualebiT daSemamagrebeli rgoli gadadis am dros koncentrul wriulSi.aseT daSvebaSi asaxvis SemTxveva Seiswavleboda m. Seremetievis(1949) mier. romelsac moyvanili aqvs dawvrilebiTi amonaxsni,ricxviTi SedegebiT, Tanafokusuri elifsuri rgolis saxismqone naxvretisaTvis. g.savinis naxseneb monografiaSi, moyvaniliagamoTvlebis Sedegebi drekadi Semagrebebis (109) asaxviTmotanili sxva formebisTvisac da naxvretis Semagrebulkonturebze Zabvebi Sedarebulia imave ZabvebTan or zRvrulmdgomareobaSi, roca Semamagrebeli rgoli absoluturadmoqnili (sicariele) da roca igi absoluturad xistia.i. aramanoviCma (1955), anviTarebda ra d. Sermanis meTodsZabvebis Sesaxeb, wriuli naxvretis mqone naxevarsibrtyeSi aago119

amocanis amonaxsni naxvretiT, romelic Semagrebulia sxvamasalis drekadi rgoliT. garemos datvirTva aq SeiZlebaganxorcieldes sxvadasxva xerxiT, magaliTad misi gaWimviT,CarCiluli rgolis Sida konturze normaluri dawneviT,sworxazovan sazRvarze Seyursuli ZaliT da sxva. amoxsnis sqemaigivea, rogorc adre (gantolebiT usasrulo sistemebze dayvana).dadgenilia, rom aq miRebuli gantolebaTa sistemakvaziregularulia naxvretis naxevarsibrtyis kidesTan nebismierisiaxlovisas.ganxiluli saxis amocanebis efeqtur amoxsnis mimarTgamoiyeneboda aseve funqciaTa wrfivi SeuRlebis meTodi.magaliTis saxiT mivuTiToT i.prusovis naSromze (1957), romelmacganixila usasrulo firfitaSi naxvretis gamagrebis Sesaxebamocana, roca cvladi kveTis mqone rgoli, SemosazRvruligaredan wrewiriT da Signidan elifsiT amagrebs mas.aqamde Cven vgulisxmobdiT, rom drekadi rgolis, romelicamagrebs firfitaSi naxvretis kides, daZabuli mdgomareoba,aiwereba iseve rogorc TviT firfitis daZabuli mdgomareoba,brtyeli drekadobis Teoriis gantolebebiT an Txelifirfitebis Runvis gantolebebiT. Tu Semamagrebeli rgolisakmarisad Txelia an aqvs fasonuri profili, maSin igi undaganvixiloT rogorc mrude Zeli, romlis deformacia aRiwerebamasalaTa gamZleobis Teoriis elementaruli gantolebebiT.aseT dasmaSi amocana Semavsebeli kideebis Sesaxebpirvelad ganixileboda m. Seremetievis (1960) mier. mudmivi kveTisSemamagrebeli rgoli miRebuli iyo wvril Rerod, romelsacgaaCnda sixiste Runvaze da gaWimvaze brtyeli daZabulimdgomareobisas da sixiste Runvaze da grexaze Txelifirfitebis Runvisas.garkveulobisaTvis ganvixiloT erTi Semagrebulinaxvretis mqone usasrulo firfitebi.Semagrebeli naxvretis konturze sasazRvro pirobebsmiviRebT, Tu moviTxovT orive mxridan Sesabamisi Zalvebisa da120

gadaadgilebebis tolobas. es pirobebi wina SemTxvevaSiwarmodgenili iyo (111) da (112) tolobebis saxiT. amjeradmiTiTebuli tolobebis marjvena nawilebSi ϕ kda ψ kfunqciebis(ukve qreba am funqciaTa ganxilvis saWiroeba) sasazRvromniSvnelobebis nacvlad gamoCndeba sxva ucnobebi, da saxeldobrgare ZalvebiXY0 n,n0, romlebic moqmedeben rgolze firfitismxridan, da rgolis RerZis gadaadgilebebi u 0, v0.Tu axla, gamomdinare mrudwiruli Reroebis mciredeformaciebis Teoriis cnobili gantolebebidan, gamovsaxavT u 0da v 0 gadaadgilebebsgareX Y datvirTvebiT da CavsvamT0 n,n0Sesabamis mniSvnelobebs zemoT naxseneb SeuRlebis sasazRvropirobaSi, maSin ϕ da ψ funqciebis, romlebic holomorfuliafirfitis areSi, gansazRvrisaTvis miviRebT or kompleqsurpirobas, romlebic marjvena nawilebSi Seicaven marto ucnobX0 nda0YnZalvebs. miTiTebuli saxis Semagrebis mqone firfitebisRunvis amocanebisaTvis marjvena nawilSi ucnobi funqciebisaerTod SegviZlia gamovricxoT, da Cven gveqneba mxolod erTisasazRvro piroba, marTalia ramdenadme ufro rTuli, vidreZiriTadi brtyeli amocanis Cveulebrivi piroba.saboloo angariSSi Cndeba amocanis efeqturi ganxilvisSesaZlebloba. naxvretebis kerZo saxeebisas, wriuli naxvretisSemTxveva emorCileba xarisxovani mwkrivebis meTodiTdawvrilebiT analizs (m. Seremetievi, 1960). arawrfiulinaxvretebisaTvis amocana ufri rTulia da efeqturi amoxsnaTxoulobs mimdevrobiTi miaxloebis meTodis gamoyenebas.am midgomis Semdgomi ganzogadeba iyo mocemuli g. savinisada b.fleiSmanis (1941) mier. miiCnevdnen ra Semamagrebel Reroswvrilad (anu Tvlidnen ra Reros ganivkveTs viwrod), maT feniskonturze sasazRvro piroba Seasustes da Camoayalibeskompleqsuri cvladis terminebSi gaerTianebuli amocanaSerbilebuli sasazRvro pirobebis mqone rgoluri Semagrebebis121

Sesaxeb. am pirobebis gamoyvanisas gamoiyeneboda daSveba imisSesaxeb, rom Rero brtyeli daZabuli mdgomareobis SemTxvevaSiRunvas ar ewinaaRmdegeba, xolo firfitebis ganivi Runvisasdakarguli aqvs grexiTi sixiste.analizuri funqciis Teoriis miRebuli amocana uSvebs,ZiriTadi brtyeli amocanebis msgavsad, amoxsnas Caketili saxiT,Tu firfitis are konformulad aisaxeba wreze racionalurifunqciis saSualebiT. es ilustrirdeba usasrulo firfitaSielifsuri naxvretis magaliTze.g. savinma da n. fleiSmanma (1964), agreTve m. Seremetievma(1960) ganixiles firfitis gaZliereba misi ganivi Runvisas sxvamasalis wvrili rgolebiT (sixistis wiboebi), romlebicganlagebulia firfitis SigniT. erTi wibos umartivesSemTxvevaSi gvaqvs Semdegi suraTi. wvrili mrudwiruli rgoli(ufro zustad, Sekruli drekadi wiri) mirCilulia firfitasTanmis Sida nawilSi. are, romelic firfitis Sua sibrtyes ukavia,iyofa am dros rgolis RerZis xaziT or bmul nawilad (Sida dagareTa am xazis mimarT). TiToeul am areSi saWiroa ganvsazRvroTkompleqsuri cvladis holomorfuli funqciebis wyvili firfitiskonturze, da agreTve rgolis xazze raime pirobebiT. SeuRlebispiroba am xazze unda SevadginoT firfitisa da Semamagrebelirgolis erToblivi muSaobis gaTvaliswinebiT (aseTi piroba samia)saboloo angariSSi oTxi holomorfuli funqciisgansazRvrisaTvis gvaqvs oTxi (110)-(112) saxis kompleqsuri piroba,romlebic Seicaven, garda mocemuli sidideebisa, rgolis RerZisrkalis orkompleqsur funqcias, romlebic ar aris mocemuliwinaswar. am gziT sixistis wiboebiT firfitis gaZlierebis Sesaxebamocana Seiswavleboda rig SemTxvevebSi. magaliTad, mrudwiruliwiboebis nebismieri ricxvis mqone mrgvali firfitisaTvisagebulia fredholmis gadawyvetadi integraluri gantoleba.zogierTi kerZo amocanebi (magaliTad, koncentrirebulisixistis wibos mqone mrgvali firfita, centraluri sixistis122

wibos mqone elifsuri firfita) ixsneba efeqturad mwkrivebismeTodiT.zemoT naxseneb g. savinis (1951), d. vainbergis (1952), m.Seremetievis (1960) da g. savinisa da n. fleiSmanis (1964)monografiebSi ganxilulia agreTve zogierTi sxva amocana,brtyeli daZabuli-mdgomareobisa da firfitebis Runvis Sesaxeb,rogorc izotropul, ise anizotropul SemTxvevebSi.gansakuTrebiT vrcladaa Seswavlili, magaliTad, sakiTxebi,romlebic dakavSirebulia anizotropuli masalis gavlenasTanelifsuri naxvretebis siaxloves Zabvebs koncentraciaze,Semamagrebeli elementebis parametrebis racionalur SerCevisSesaxeb, konturuli Seyursuli datvirTvebis mravalfenovandiskSi gavlenis Sesaxeb.n.musxeliSvilis ideebma didi gavlena moaxdines analizurfunqciaTa Teoriis sasazRvro amocanaTa da singularulintegralur gantolebaTa Teoriis problematikur damuSavebaze(T. gaxovis, i. vekuas, n. vekuas, a.biwaZis, d. kveselavas, b.xvedeliZis, l. maRnaraZis, g. manjaviZis da sxv. naSromebSi).igive ideebi ukanaskneli sami aTeuli wlis manZilzemtkiced ikidebs fexs elifsuri tipis kerZowarmoebulebiandiferencialur gantolebaTa zogad TeoriaSi (i. vekuas, z.xalilovis da sxv. naSromebi). kerZod, maT arsebiTi gamoyenebahpoves Txelkedliani sivrciTi sistemebis Teoriis sakiTxebSi.aRsaniSnavia l. muxaZisa da g. yifianis mimoxilviTixasiaTis statia `saqarTveloSi Txelkedliani sivrciTisistemebis Teoriis gamokvlevaTa mimoxilva~ (mecniereba dateqnika # 4-5. 1997).2.3.3. Sereuli da sakontaqto amocanebiSereuli da sakontaqto amocanebi ganekuTvnebiandrekadobis Teoriis yvelaze rTul amocanaTa ricxvs.kompleqsur cvladis funqciaTa Teoriis meTodebiT maTi123

Seswavlisas miiReba sasazRvro pirobebi wyvetis koeficientebiTda Cndeba wyvetis wertilebis midamoebSi amonaxsnebis Seswavlisaucilebloba.rogorc zemoT ukve aRiniSna (94) d. Sermanma (1940) aagowyvetis koeficientebis mqone singularuli integralurigantoleba ZiriTadi Sereuli brtyeli amocanisaTvis: igivegantoleba iZleva normalurad datvirTuli Txelianizotropuli firfitis Runvis amocanis amoxsnis saSualebas,roca kides nawili Camagrebulia, xolo nawili – Tavisufali.a. kalandiam (1952) aago singularul integralurgantolebaTa sistema firfitebis Runvis zogadi amocanisamoxsnisaTvis, roca kides nawili Camagrebulia, nawili –dayrdnobili, xolo danarCeni – Tavisufali. rig naSromebSi (ix.,magaliTad, a. kalandia, 1961; d. Sermani, 1955) mocemulia firfitebisRunvis Sereuli amocanebis ricxviTi amonaxsni kerZo saxisareebisaTvis.brtyeli drekadobis Teoriis Sereuli amocanebis amoxsniserT-erT yvelaze efeqtur meTods warmoadgens funqciaTa wrfiviSeuRlebis meTodi. am meTodiT Sereuli amocanebis amoxsnisSesaxeb iyo Tqmuli zemoT (97).drekad naxevarsibrtyeSi xisti tvifarebis Cawnexvis Sesaxebamocanebs mivyavarT SeuRlebis sasazRvro amocanebamde,romlebic analogiuria ZiriTadi Sereuli amocanisaTvis zemoTagebuli (97) SeuRlebis amocanisa. ori drekadi sxeulisurTierTSexebis amocana (hercis ganzogadebuli brtyeliamocana), romlebic formiT axlosaa naxevarsibrtyesTan, rocaurTierTSexebis ubani mcirea, aseve daiyvaneba wrfivi SeuRlebisamocanaze. am amocanebis amoxsna funqciaTa wrfivi SeuRlebismeTodiT moyvanilia n.musxeliSvilis monografiaSi.l. galinma (1953) mogvca rigi sakontaqto amocanebisamoxsnebi kompleqsuri cvladis funqciis Teoriis gamoyenebissaSualebiT. i.Staermani (1949) Seiswavlida sakontaqto amocanebsintegralur gantolebaTa meTodiT.124

v. abramovis (1937), n. glagolevis (1942, 1943), v. mosakovskisda p.zagubiJenkos (1955), v. panasiukis (1953, 1954), a. kalandias (1957,1958), m. Seremetievis (1952, 1961) naSromebSi kontaqtur amocanaTarigi gamokvleulia sxvadasxva meTodebiT.2.3.4. anizotropuli tanis drekadobis Teoriisbrtyeli statikuri amocanakompleqsuri cvladis funqciis Teoriis meTodebi, rogorcpirvelad gvaCvena s. lexnickim, warmatebiT SeiZleba iyosgamoyenebuli anizotropuli sxeulis brtyel amocanaSic (ammimarTulebiT s. lexnickis pirveli naSromebi iyogamoqveynebuli ocdaaTian wlebSi; ix., magaliTad, monografia: s.lexnicki, 1947, gam. 2-1957).vTqvaT erTgvarovan anizotropul sxeuls yovel wertilSigaaCnia drekadi simetriis sibrtye, paraleluri mocemulisibrtyisa, romelsac Cven miviCnevT Oxy sibtyed. Tu sxeulieqceva drekadi deformaciis qveS, romelic Oxy sibrtyisparaleluria, maSin ZabvaTa funqcia (eris funqcia) akmayofilebsganzogadebul biharmoniul gantolebas (mxedvelobaSi gvaqvsmoculobiTi Zalebis ararsebobis SemTxveva)44444∂ U ∂ U ∂ U ∂ U ∂ Ua0+ a1+ a2+ a3+ a4= 0 , (113)432 234∂x∂x∂y∂x∂y∂x∂y∂ysadaca , L a − namdvili mudmivebia, romlebic damokidebulia0,4gansaxilveli tanis drekad Tvisebebze (analogiur gantolebasaqvs adgili firfitis ganzogadebuli brtyeli daZabulimdgomareobisaTvisac).am SemTxvevaSic xerxdeba avagoT amonaxsnis zogadiwarmodgena kompleqsuri cvladis ori analizuri funqciissaSualebiT. es warmodgena damokidebulia maxasiaTebeligantolebis, romelic Seesabameba (113) gantolebas, fesvebze2 3 4a + a s + a s + a s + a s 0 . (114)0 1 2 3 4=125

ogorc s. lexnickim gviCvena, am gantolebas ara aqvsnamdvili fesvebi. izotropuli tanis SemTxvevaSi (114) gantoleba2 4daiyvaneba 1+ 2s + s = 0 gantolebamde da maSasadame, aqvsorjeradi i da _i fesvebi. Tu (114) gantolebas aqvs orjeradifesvebi s = α + iβ, s = α − iβ, maSin (113) gantolebis zogadinamdvili amonaxsni warmogvidgeba saxiTU ( x,y)= zϕ ( z)+ zϕ(z)+ x(z)+ x(z).(115)rogorc izotropuli tanis SemTxvevaSi, magram amjeradkompleqsur cvlads z-s aqvs saxe z= x + sy = x + α y + iβy(( x,y)∈ s),sadac s aRniSnavs ares, romelic ukavia tans.movaxdenT ra afinur gardaqmnebsx′ = x + αy, y′ = βy. (116)mivalT kompleqsur cvladTan z ′ = x′+ iy′, romelic icvlebas′ areSi da miiReba s aredan (116) afinuri gardaqmnebissaSualebiT.(115) formula da misgan gamomdinare gamosaxuleba Zabvebisada gadaadgilebebis komponentebisaTvis, gviCveneben, rom esSemTxveva (anu (114) gantolebis jeradi fesvebis SemTxveva)warmoadgens TiTqmis srul analogias izotropuli tanisSemTxvevisa, da amitom mas Cveulebriv gamoricxaven ganxilvidan.SemTxvevaSi, roca (114) gantolebas ara aqvs jeradi fesvebianu aqvs oTxi sxvadasxva wyvilebad SeuRlebuli fesvebis α + iβ1= 1, s1 α1− iβ1= , s2= α2+ iβ2, s2 = α2− iβ2,(113) gantolebis zogadi namdvili amonaxsni warmogvidgeba saxiTU x,y)= F ( z ) + F ( z ) + F ( z ) + F ( ) , (17)(1 1 1 1 2 2 2z2ori analizuri funqciis saSualebiT, sadac cvladebiaz= x + s y = x + α y + iβ, = x + s y = x + α y + i y1 11 1yz2 22β2romlebic Sesabamisad icvlebian s 1 da s 2 areebSi da miiRebian saredan Sesabamisi afinuri gardaqmnebiT.gansaxilvel SemTxvevaSi, izotropuli tanis SemTxvevisagangansxvavebiT, gvixdeba saqme viqonioT or sxvadasxva z 1 da z 3126

kompleqsuri cvladis analizur funqciasTan, romlebicicvlebian or sxvadasxva areSi (advili dasanaxia, rom z 1 da z 2cvladebi dakavSirebulia erTmaneTTan afinuri da araanalizuri gardaqmnebiT). es garemoeba, unda iTqvas, arTulebssasazRvro amocanebis amoxsnas (anizotropuli tanis SemTxvevaSiefeqturad amoxsnadi amocanebis klasi gacilebiT viwroa, vidreizotropuli tanebisaTvis). Tumca anizotropuli tanisSemTxvevaSic xerxdeba sasazRvro amocanebis amoxsnis miRebakompleqsuri cvladis funqciis Teoriis meTodebis saSualebiT.rigi mniSvnelovani Sedegebisa am mimarTulebiT ekuTvniTs.lexnickis, s. mixlins, g. savins da sxva.(117) ZabvaTa funqciis zogadi warmodgenidan gamomdinareobsZabvebisa da gadaadgilebebis Semdegi kompleqsuri warmodgenebidF1aq Φ1(z1)= ;dz1XYyXx= 2 Re= 2 Rey= −2 Reu = 2 Rev = 2 ReΦ2(2)22[ s Φ′ ( z ) + s Φ′ ( z )],111⎫⎪⎬[ Φ′1(z1)+ Φ′2(z2)],[ s Φ′ ( ) + Φ′ ( )];⎪ 1 1z1s22z2⎭2[ p1Φ1(z1)+ p2Φ2(z2)] − ωy+ u0[ q Φ ( z ) + q Φ ( z )] + ωx+ v ⎭ ⎬⎫1dFz =dz2121222220(118). (119), p 1 , p 2 , q 1 , q 2 _ sruliad gansazRvrulimudmivebia, romlebic gamoisaxebian tanis drekadi konstantebiT,xolo ω, u 0 , v 0 _nebismieri (namdvili) mudmivebi Seesabamebian tanisxist gadaadgilebas.Tu s are, dakavebuli sxeulia, caladbmulia, maSinanalizuri funqciebi, romlebic monawileoben zogad kompleqsurwarmodgenebSi, calsaxebia; mravladbmuli ares SemTxvevaSi isini,mravalsaxa analizuri funqciebia. magaliTad, Tu s areSemosazRvrulia ramdenime konturiT, maSin Φ ) da Φ )funqciebs aqvT saxe1( z 12( z 2*Φ ( z ) = Φ ( z ) +Φ121*( z ) = Φ ( z21212) +n∑k = 1n∑k =⎫Akln( z1− zik),⎪⎬B ln( − ). ⎪kz2z2k1⎪⎭(120)127

**sadac Φ1( z 1) da Φ2( z 2) − calsaxa analizuri funqciebia, z 1 kdaz 2− fiqsirebuli wertilebia, xolo A k da B k mudmivebi, romlebickgamoisaxebian gare Zabvebis mTavari veqtoriT, romelic modebuliasasazRvro konturebze. Tu ZalvaTa mTavari veqtori, romelicmodebulia TiToeul konturze, tolia nulis, maSin mudmivebi A k ,B k tolia nulisa da Φ ), Φ ) funqciebi calsaxaa S 1 , S 2areebSi.1( z 12( z 2(118) formulebidan gamomdinareobs, rom pirveli ZiriTadiamocanis amonaxsni (sazRvarze mocemulia gare Zalvebi X n , Y n )daiyvaneba S 1 da S 2 areebSi analizur Φ ) da Φ ) funqciebispovnaze sasazRvro pirobiTsadacc ,c 1 22 Re2 Re[ Φ ( z ) + Φ ( z )]= −[ s Φ ( z ) + s Φ ( z )]1111122222s∫0Y d=ns∫0X1( z 1s+ c ,− nebismieri namdvili mudmivebi.nds1+ c2⎫⎪⎬, ⎪⎪⎭2( z 2(121)analogiurad, meore ZiriTadi amocana daiyvaneba Φ ) da1( z 1Φ 2( z 2) analitikuri funqciebis moZebnamde sasazRvro pirobebiT2Re2Re[ p1Φ1(z1)+ p2Φ2( z2)] −ωy+ u0= g1,[ q Φ ( z ) + q Φ ( z )] + ωx+ v = g . ⎭ ⎬⎫11122202(122)sadac q 1 da q 2 _ u da v gadaadgilebaTa komponentebis mocemulisasazRvro mniSvnelobebia.ZiriTadi Sereuli amocanis SemTxvevaSi gvmarTebs movaxdinoT(121) da (122) pirobebis kombinireba, zogad SemTxvevaSi miviRebTsasazRvro amocanas wyvetis koeficientebiT.saWiroa aRiniSnos, rom Txeli anizotropuli firfitisRunvis amocana, iseve rogorc izotropuli firfitis SemTxvevaSi,daiyvaneba kompleqsuri cvladis funqciis Teoriis amocanebze,romlebTanac mivyavarT brtyel amocanebs. Camagrebuli kidesmqone anizotropuli firfitis Runvis amocana daiyvaneba (121)saxis amocanaze, xolo Tavisufali kides mqone firfitis Runvisamocana (122) saxis amocanaze. nawilobriv Tavisufali da128

nawilobriv Camagrebuli kides mqone firfitis Runvis Sereuliamocana daiyvaneba brtyeli Teoriis ZiriTadi Sereuli saxisamocanaze (ix. s.lexnicki, 1947).saWiroa aRiniSnos, rom zogierT stacionarul dinamikuramocanasac mivyavarT zemoT miTiTebul amocanebamde ase, l. galinma(1953), ganixilavda ra amocanas tvifaris Sesaxeb, romelic moZraobdamudmivi siCqariT izotropuli naxevarsibrtyis sazRvris gaswvriv,daiyvana is kompleqsuri cvladis funqciis Teoriis sasazRvroamocanamde da am gziT aago misi amonaxsni.2.3.5. sivrciTi maRali Zabvis eleqtrogadamcemixazebis konstruqciebis qarsawinaaRmdegomdgradobaeleqtrogadamcemi xazebis xangrZlivobis problemamsoflios mravali qveynisaTvis aris mniSvnelovani. kavkasiisregioni rTuli klimaturi da mikroklimaturi pirobebiTgamoirCeva da misi cvalebadobis Sesaxeb informaciiT ar arisuzrunvelyofili, radgan mTianeTSi ganlagebuli mravalisadguri, romelTa Soris manZili xSirad 40 km-s aRemateba, veruzrunvelyofs sando meteoladastris Sedgenas.wlebis ganmavlobaSi saqarTvelos teritoriazeqariSxlebis moqmedebis Sedegad adgili hqonda eleqtroavariebsgadamcem xazebze `kavkasioni~, `imereTi~, `paliastomi~, `kolxida~da sxv. am SemTxvevaSi umetes wilad ziandeba Sualedi sayrdeniТ-2б П24Ма. sayrdenis dazianebisas irRveva qveda panelebi dasayrdeni miwaze vardeba. arc erT SemTxvevaSi ar momxdaraelementebis amogleja sayrdenebis saZirkvlebidan. mrRvevqariSxlebs adgili hqonda 1987 wlis oqtomberSi, 1988 wlisTeberval-martSi da a.S.eleqtrogadamcemi xazebi daproeqtebuli iyo qarissiCqarisaTvis 36 m/wm-Si, qaris siCqariT dawnevisaTvis 80 kgZ/m 2 .saangariSo datvirTvebis ganmeorebadoba – 15 weliwadSi erTi.129

sinamdvileSi qaris siCqarem gadaaWarba 40 m/wm-Si. qarissiCqariTi dawneva 80 kgZ/m 2 -ze meti aRmoCnda. ganmeorebadoba, kiSeadgina 1→10 weliwadSi.dainteresebuli mxaris TxovniT SeviswavleT BΛ-500kilovatiani `imereTis~ xazis avariuli situacia # 317-339sayrdenebs Soris. kompiuteruli programebis SedgenisasgaTvaliswinebul iqna lipyinulebisa da qariSxlebiserTdrouli moqmedeba, agreTve samTo reliefis Taviseburebani.qariSxali saqarTvelos mTian regionebSi rogorc wesi 3-10dRe-Rame grZeldeba da xasiaTdeba qaris simZlavriscvalebadobis didi amplitudiTa da impulsuri qmedebiT. undaaRiniSnos, rom sayrdenebis rxevebi aseve xasiaTdeba didiamplitudebiT, rasac Tan axlavs konstruqciebis daRlilobismovlenebi. sayrdenebis rxevaTa did amplitudebs adgili hqondarogorc Runvis, aseve grexis deformaciebisTvis.mniSvnelovani RunviTi rxevebisTvis, dalamberis principisgamoyenebis safuZvelze erTmaliani sayrdenisTvism y ′′ = − f ( y)+ p , (123)sadac f (y)aRmdgeni Zalaa; p _ statikuri datvirTva; m _ masa;f(y) SeiZleba warmovadginoT (Seferis mixedviT) teilorismwkrivis saxiT:3y − y0( y − y0) ( y − y0)f ( y)= f ( y0) + f ′(y0) + f ′′(y0) + f ′′′ ( y0) , (124)1!2!3!sadac y 0statikuri CaRunvaa; y _ CaRunva nebismier kveTSi. (124)-Si SesaZlebelia CaiTvalos, romy − y 0≈ y , ris Sedegadac23y yf ( y)= f ( y0) + f ′(y0) ⋅ y + f ′′ ( y0) + f ′′′( y0) . (125)2! 3!praqtika gviCvenebs, rom SesaZlebelia pirvel sam wevrsdaveyrdnoT, miviRebTsadac2 2m ⋅ y′′+ a y + by = 0.(126)2a = f ′( y 0); b = f ′′ y ) da f ( y 0) = p.( 0130

(125)-is amonaxsni warmovadginoT funqciuri mwkrivis saxiTy2 3= ϕ0( t)+ ϕ1(t)⋅b+ ϕ2( t)b + ϕ3b. (127)sawyisi pirobebis gaTvaliswinebiT miiRebaϕ ( t)= ya cosωt(128)0⋅sadac ω =2a.mgardaqmnebis Sedegad miviRebT arawrfivi rxevebisgantolebas2⎛ y ⎞⎡yyy( t)= ⎜1+0,667 ⋅⎟⎢−0.5⋅0+ y cosωt+ 0,166 cos 2ωt+22 02⎝ a ⎠⎣aa3y ⎤0+ 0.041⋅⋅ cos3ωt2 ⎥ .a ⎦(129)(129) rxevebis RunviTi parametrebis gamoTvlis saSualebasiZleva. msgavsi analogiiT SeiZleba dadgindes sayrdenis grexiTirxevebis parametrebic:23⎛ ϕ ⎞⎡ϕϕϕ( t)= ⎜1+0,667 ⋅00⎟⎢−0.5 ⋅ + ϕ cosωt+ 0,166 cos 2ωt+22 02⎝ a ⎠⎣aaϕ0⎤+ 0.041⋅⋅ cos3ωt2a⎥ ,⎦(130)sadac1ω = ,MM _ mgrexi momentia da Mmr2= ⋅ .eleqtrogadamcemi xazebis sayrdenebis rRvevis albaTobisgamomJRavneba aqtualuri sakiTxia, radgan maszea damokidebulieleqtrogadamcemi xazebis muSaobis resursi.praqtikam gviCvena, rom am xazebis daproeqteba arsebulinormebis mixedviT ver uzrunvelyofs mis muSaobas 10-15 wlisganmavlobaSi; sayrdenebis konstruqciebSi dabali sixSirisvibraciuli daRlilobis Sedegad xdeba dazianebaTa dagroveba,amasTan, daRlilobis gamomwvevi faqtorebi albaTobiT xasiaTsamJavnebs.131

aRmoCnda, rom eleqtrogadamcemi xazebis rRvevismniSvnelovan risks warmoadgens garsis da sayrdenebissartylebis SemaerTebeli kvanZebis daRliloba, ris Sedegadacsayrdenebis rxevebis Runva-grexiTi amplitudebi, zRvruliamplitudebis N ricxvi (ciklebis raodenoba) σa da σm ciklisparametrebTan mimarTebaSi ganisazRvreba gamosaxulebiT⎛ σ −1⎞N = N0⎜ ⎟ ,⎝σa+ ϕσm⎠sadac N 0 ciklTa bazuri sididea, roca2σ−1−σ0σm≥ 0 da σmax≤ [ σ den],ϕ =σϕ _ koeficienti gansazRvravs saSualo ZabvaTa gavlenas ciklTazRvrul raodenobaze rRvevamde.dazianebaTa dagrovebis siCqare konstruqciaSi SeiZlebawarmodgenil iqnes ori funqciis namravliT, romelTagan erTidamokidebulia Zabvebis sididesa da niSanze, meore ki –dazianebaTa sidideze:∂v∂v(σ ) ∂v(n)= ⋅ = f1(σ ) ⋅ f2( n),∂n∂n∂nsadac v dazianebis parametria (lodes parametri); v = 0, rocanageboba 100%-is dauzianebelia da v = 1, roca nageboba mTlianaddarRveulia. SeiZleba Caiweros, rom1∫∂ v)f ( v(nn∂n= ∫ =fm( ] ∫σ ( σ ]0 2) 0 1 0sadac n ciklTa raodenobaa saTanado σ rxevebisaTvis. TudnNdatvirTvis reJimi mravalsafexuriania, maSinK∑i=1niNi1= 1 = ∫ dv = 1;0aq nλ ciklTa raodenobaa σi ZabvebisaTvis; Ni-zRvrulciklTa raodenoba igive σi ZabvebisaTvis; K _ datvirTvisseriaTa ricxvi. saSiSi Zabvebis gaCenis kriteriumad iTvlebadaRlilobis ZabvaTa intensiurobis K d koeficienti:,0132

Kni( λ , n ) ⋅ σi⋅ f l ,Ki bzNsadac f SenaduRi nakeris fardobiTi sigrZe an moqlonTad= fSed moq ∑i=1diametrisa da raodenobis gavlenis funqciaa;Kn∑σiii= 1 N− rRvevispirobebis integraluri forma ciklTa raodenobis mixedviT i-ursafexurze; K-datvirTvaTa seriebis raodenoba, l bz -bzarebissigrZe; f i -ciklTa asimetriis maCvenebeli i-ur ciklSi.sayrdenTa rRveva xasiaTdeba ganviTarebis sami zoniT:a) latenturi rRvevis zona, sadac kvlevis zusti meTodebigamoavlens procesis dasawyiss.b) rRvevis zomieri siCqaris zona Tavs iCens rRvevisxiluli Sedegebi: saWiroa efeqturi zomebis miRebakonstruqciebis remontisa da gaZlierebisaTvis.g) rRvevis siCqaris katastrofuli sidide. es avariulisituaciaa, romlis aRkveTa praqtikulad SeuZlebelia.qaris zemoqmedeba aRiqmeba, rogorc arastacionaruli,SemTxveviTi qmedeba. aseT SemTxvevaSi statikurad rRvevadi dageometriulad konformuli sayrdenebisaTvis simtkicis pirobisSesrulebis albaToba[ σ ≤ ]H = Π ⋅ H = Π P ,c=mi=mi= 1 i i=1 iσ*sadac m-kvazi damokidebul elementTa (panelTa) raodenobaasayrdenis konstruqciaSi.konstruqciis mtyuneba simtkicis TvalsazrisiT SesaZlebeliamoxdes im SemTxvevaSi, rodesac datvirTvis realuri procesiiwvevs σ*-gadametZabvas. mtyunebas SeiZleba hrondes adgiliagreTve dazianebaTa dagrovebis saSiSi akumulaciis dros, rocaK*xdeba kritikulze meti.post-faqtum avariuli mdgomareobis Seswavlisas, romelicqaris moqmedebiT aris gamowveuli, mizanSewonilia miaxloebiTi,magram sakmaod sando meTodebis gamoyeneba saangariSo Zabvebis133

gansazRvrisaTvis daRlilobis akumulirebuli SedegebismxedvelobaSi miRebiT.saSiSi kveTis (kveTebis) dadgena eleqtrogadamcemi xazebissayrden elementebSi unda moxdes imis gaTvaliswinebiT, rom aseTSemTxvevaSi adgili aqvs ganiv Runvas (nebismieri mimarTulebiT)da grexas grZivi RerZis mimarT. amasTan, kuTxis sayrdenebSi(kuTxis elementebSi) Runva iwvevs gaWimvas an kumSvas. undagaviTvaliswinoT isic, rom grexis didi deformaciebis gamoeleqtrogadamcemi xazebis sayrdenebis grZiv elementebSi gveqnebaRunvis deformaciac.yvela am SemTxvevaSi gisosis elementebi ganicdian grZivgaWimvas an kumSvas.yovelive zemoaRniSnulis gaTvaliswinebiT saangariSo Zabvaσ i = A⋅σ+ Bσ+ Cτxyxy2 − λ 2 − λ λ −1A = + cos 2α+ , sin 2α;2 222 − λ 2 − λ λ −1B = − cos 2α− , sin 2α;2 22C = ( λ −1)cos 2α− (2 − λ)sin2α.aq λ koeficientia, romelic masalis mgrZnobelobas asaxavsnormaluri da mxebi Zabvebis zemoqmedebisas simetriul anσ −1mapulsirebel ciklSi λ = . τ −1maqsimaluri mxebi Zabvebis hipoTezis gamoyenebis SemTxvevaSiτ = − 10,5 σ da λ = 2 .−1maqsimaluri oqtaedruli Zabvebis hipoTeziTσ da λ = 1, 43 .0,7= τ − 1 −1idealurad plastikuri masalebisaTvis λ=2; myifemasalebisaTvis λ = 1. eleqtrogadamcemi xazebis sayrdenebisaTvisλ = 1,43.aucilebelia moviyvanoT zogierTi mosazrebaeleqtrogadamcemi xazebis sayrdenebis gaangariSebis TaviseburebaTaSesaxeb.134

1. qaris zemoqmedebis gavlena, nagebobaTa sakuTar rxevebTanSedarebiT, ufro mcirea, vidre mikrometeorologiuri pikissidide vanderxovenis mixedviT.2. moqnili sistemis (vanturi xidebi, maRlivi nagebobebi,eleqtrogadamcemi xazebis sayrdenebi) sakuTari rxevebi xvdebaqaris speqtris sagrZnoblad didi ordinatebis zonaSi.1-l naxazze naCvenebia zogierT nagebobaTa sakuTarirxevebis intervalTa miaxloebiTi sidideebi.nax. 1. nagebobaTa sakuTari rxevebis intervalTamiaxloebiTi sidideebivertikalur RerZze gadazomilia fardobiTi speqtrulisixSiris simkvriveebi, horizontalurze-sixSireebi. rogorcgrafikidan Cans, eleqtrogadamcemi xazebis sakuTari rxevebissixSire icvleba 0,1÷1 hercis farglebSi, sakmaod maRalispeqtruli simkvrivisaTvis – 0,4 da meti.gairkva, rom mTagorian raionebSi eleqtrogadamcemi xazebisdaproeqtebisas gaTvaliswinebuli ar aris qaris siCqaris didicvalebadoba borcvebis zonaSi. arada, rogorc es qvemoTnaCvenebi naxazidan (nax. 2) Cans ar Seesabameba saproeqtomocemulobas. qaris siCqare borcvebis zeda zonaSi daaxloebiT50%-iT izrdeba.wyvetilebiT naCvenebia qaris siCqaris epiura, romelicsafuZvlad udevs eleqtrogadamcemi xazebis sayrdenebis135

gaangariSebas daproeqtebis procesSi, xolo uwyveti xaziTnaCvenebia qaris siCqaris realuri epiura.nax. 2. qaris siCqaris saangariSo da realuri epiurebiam movlenis gauTvaliswinebloba eleqtrogadamcemi xazebissayrdenebisaTvis, romlebic ganlagebulia borcvebis uSualosiaxloves, iwvevs qaris Cqarosnuli dawnevis saangariSo sididisgaumarTlebel Semcirebas, rac am eleqtrogadamcemi xazebissayrdenebis avariebis erT-erTi mTavari mizezia.qaris datvirTvis normatiuli dinamikuri Semdgeni,romelic miRebulia i-ur ubanze, roca sayrdeni irRveva i-uriformiT, ganisazRvreba formuliTQPQRij= M η ⋅ξνjijisadacGiMj= i-uri ubnis masaa,gηij − i -ur ubanze moqmedi aCqareba;ξ − dinamikurobis koeficientia.iV _koeficienti, romelic gansazRvravs qaris siCqarispulsaciis sivrciT korelacias sayrdenis simaRlesTandakavSirebiT. dinamikurobis koeficientiT mxedvelobaSi miiRebanagebobebis sixSiris amplitudis zrda, roca nagebobebzemoqmedebs gare Zalebi, romelTa sixSire emTxveva sistemissakuTar sixSireebs rezonansul reJimSi.ξ-gansazRvrisaTvis Cven miviReT empiriuli gamosaxuleba136

λ1,10ξ = 3 + ln .265es formula miRebulia eqsperimentuli mrudebis damuSavebis(129) Sedegad foladis konstruqciebisaTvis. formulis sizustegarantirebulia 5-7%-is farglebSi. ξ-simbolo gamosaxulebasaqvs Semdegi saxe:ξ =3 + ln( 4 ⋅Ti⋅ n ⋅ )sadac q 0qaris Cqarosnuli dawnevaa ubanze, niutonis mixedviT:q02ν=16Ti − i -uri formis rxevaTa periodi; n – gadatvirTvis koeficienti.265obieqturi monacemebis safuZvelze damuSavda programa.dadgenilia, rom eleqtrogadamcemi xazebis sayrdenebis avariashqonda adgili qariSxlis Sedegad, roca qaris siCqareaRemateboda 45 m/wm-Si. rRvevis momentisaTvis Zabvebma saSiSkveTSi gadaaWarba normatiul maCvenebels fol 3 markisTvis.q 01.12.3.6. bzarebis mqone myifi sxeulebiswonasworobamyife masalebis rRvevis Teoriis ageba dakavSirebuliadrekad sxeulSi gadaadgilebaTa velis wyvetis zedapirismidamoSi (`bzarebSi~) daZabuli mdgomareobis SeswavlasTan.yvelaze martivs warmoadgens amocana brtyeli daZabulimdgomareobis Sesaxeb wrfivi Wrilis mqone filaSi, romelicdatvirTulia ZalebiT, perpendikularuliT Wrilisadmi, romlisboloebi sakmaod dacilebulia filis kideebisagan. wrfivdasmaSi klasikur amonaxsns, romelic miiReba zRvruligadasvliT elifsuri naxvretis midamoSi daZabuli mdgomareobisSesaxeb amocanis amonaxsnidan, mivyavarT usasrulo Zabvebamdebzarebis boloebSi (ares kuTxis wertilebSi, damatebiTidaSvebebis gareSe is ar iZleva saSualebebs vimsjeloT,137

datvirTvis parametris ra mniSvnelobisas daiwyebs bzarigavrcelebas – daiwyeba myife rRvevis procesi.myife rRvevis Teoria momdinareobs a. grifitisnaSromebidan (1920), romlebic gagrZelda j. irvinis (1948 daSemd.) da g. orovanis (1950 da Semd.) mier. Sedegad myife rRvevazemasalis simtkicis maxasiaTeblebisaTvis Semotanili iqna axalikonstanta, romelic iZleva saSualebas ganvixiloT myifebzarebis Sesaxeb amocana drekadobis klasikuri Teoriis dasmaSi.bzarebis zrdis kinetikis sakiTxebi ganixilebodag.barenblatis, v.entovis, r. salganikis (1966, 1967) da agreTve g.barenblatis, r.salganikis, g. Cerepanovis (1962) mier. l. kaCanovs(1961) ekuTvnis cda mieca Sefaseba bzaris mqone tanisxangamZleobisaTvis drekad-blant tanSi.g. barenblatma da g. Cerepanovma (1960) ganixiles amocana,orTotropuli drekadi tanis gapobis Sesaxeb Txeli xistisoliT, romelic gadaadgildeboda Tanabari siCqariT. usasrulotanis sasrulo sigrZis soliT gapobis Sesaxeb amocanaSi i.marzukovma (1961) miiRo bzaris sigrZis damokidebuleba solissigrZisagan. Zvris bzarebis gavrceleba ganixiles g. barenblatmada g. Cerepanovma (1961). sixistis wiboebiT Semagrebuli bzarebisganviTarebis mdgradobis Sesaxeb amocana ganixileboda e.morozovasa da v. partonas naSromSi (1961). bzarebisormagperioduli sistemis mdgradi ganviTareba v. partonam (1965).g. Cerepanovma (1966) Seiswavla bzarebis ganviTareba SekumSultanebSi.bzaris modeli, romelSic gaiTvaliswineba aseveSeWidulobis Zalebi ubnebze, romlebic Tanazomadia bzarissigrZesTan, ganixileboda, bzaris kideebis mdore daxurvisa damaTze Zabvebis sasruloobis pirobebis gamoyenebiT, m. leonovisada v. panasiukis (1959) mier. mocemulia amonaxsnebi brtyeliamocanebis didi ricxvisa, sxeulis zRvruli wonasworobisSesaxeb bzarebis sxvadasxvagvari ganlagebisa da formisas,bzarebis mqone sxeulis sxvadasxvagvari xerxiT datvirTvisas138

(v.panasiuki da d. bereJnicki, 1964-1966). amave klass miekuTvnebabrtyeli amocanebi, naxvretis konturis kuTxis wertilebismidamoebSi daZabuli mdgomareobis Sesaxeb (v. panasiuki da g.buini, 1966). kerZod radialuri bzarebis mqone wris Sesaxeb(v.panasiuki, 1965).usasrulo marTkuTxa amonaWeris, romlis fskersac awvebaxisti tvifari, kuTxis wertilebidan bzarebis dawyebiTiganviTareba, Seswavlilia g. Cerepanovis (1963) mier.bzaris kidis siaxloves daZabuli mdgomareobis Sesaxebamocanebi, roca is gamodis filis kideze an masTan axloa,ganixileboda v.panasiukis (1960), g. barenblatis da g. Cerepanovis(1960, 1962) mier. amocana gadatvirTvebis (mRunavi momentis,Tanabradganawilebuli wnevis) zRvruli mniSvnelobebis Sesaxebzolze (koWze), romelsac sworxazovani bzari aqvs zolisRerZis marTobulad, ganixileboda b. lozovois da v. panasiukis(1961-1963) mier. gegmaSi wris formis mqone brtyeli bzariT taniszRvruli wonasworobis Sesaxeb sivrciTi amocanebi ganixiles m.leonovma da v. panasiukma (1961). elifsuri bzaris ufro rTuliSemTxveva ganixileboda v. panasiukis (1962), m. leonovisa da k.rusinkos (1963, 1964) mier.2.3.7. myife da plastikuri masalebis rRvevisproblemebiistoriulad Camoyalibda inJinrebis Sexeduleba masalissimyifesa da plastikurobaze imis mixedviT, Tu rogor iqceva esaTu is masala laboratoriuli gamocdis dros martivi gaWimvisSemTxvevaSi. ase magaliTad, Tuji iTvleba myifed imis gamo, rommisgan damzadebuli nimuSi gaWimvis Sedegad irRveva mciredeformaciis pirobebSic ki, amasTan bzarebi zedapirze lagdebagrZivi RerZis marTobuli mimarTulebiT. am movlenas Semdegiaxsna aqvs: Tujis SemadgenlobaSi Sedis Tavisufali grafiti,ris Sedegadac Tuji emsgavseba myife araliTonuri bunebis mqonemasalas. Tu amave wesiT gamovcdiT rbili foladisagan139

damzadebul nimuSs, davinaxavT, rom mis zrdapirze bzarebisakmarisad mniSvnelovani sididis deformaciebis fonzeviTardeba da amboben, rom igi plastikuri masalaa. amasTanave,Tu imave rbili foladisagan damzadebul nimuSs Zalze swrafaddavtvirTavT, is myife masalisaTvis damaxasiaTebel Tvisebebsgamoavlens. gaWimvaze rbili foladis gamocdis procesis dros,im momentSi, roca datvirTva miaRwevs am nimuSisaTvismaqsimalur sidides, nimuSis zedapirze garkveul kveTSiwarmoiqmneba Seviwroveba (e. w. yeli). am momentidan Rero gadadissamRerZa dabzarul mdgomareobaSi da bzarebis gaCenamdedeformacia mimdinareobs TandaTan cvladi, rTuli samRerZadaZabul-deformirebuli mdgomareobis pirobebSi.p. bridJmenma (1944), davidenkovma da n. spiridonovam (1946)aCvenes, rom samRerZa Zabvebis maqsimumi miiRweva grZiv RerZze,minimumi ki – gverdiTi zedapiris siaxloveSi. bzari jer gaCndebanimuSis grZiv RerZTan da misi perpendikularuli iqneba.mikroskopuli dakvirvebebi cxadyofs, rom rRveva moxdebamowyvetiT. gverdiTi zedapiris siaxloves bzaris forma icvleba,is ukve Zvris Sedegad viTardeba.unda aRiniSnos, rom mowyvetiT rRveva sakmaod didiplastikuri deformaciebis warmoSobis Semdeg xdeba. amrigadoTaxis temperaturis pirobebSi Catarebuli cdis SedegadTandaTanobiTi zrdadi datvirTvis pirobebSi nimuSis rRvevaxdeba mowyvetisa da Zvris Sedegad.temperaturis Semcirebis Sedegad izrdeba imis albaToba,rom nimuSis rRveva moxdeba mowyvetiT. d.parkeri, h.devisi daa.flanigani (1946) mividnen daskvnamde, rom sakmarisad dabalitemperaturis dros „fialiseburi“ bzarebi, rbili foladisagandamzadebul nimuSSi gardaiqmnen mowyvetis bzarebad, romlebicoTaxis temperaturis pirobisaTvis Tujis nimuSis zedapirzewarmoqmnili bzarebis msgavsia.mniSvnelovani kvlevebi temperaturis gavlenis Sesaxebbzarebis formaze Catarebul iqna meore msoflio omis periodSi,140

adgan xSiri iyo SeduRebuli liTonis konstruqciebis ngrevisSemTxvevebi.erTi da igive reJimis cdis procesSi, erTi da igivemeTodis gamoyenebis dros SemCneuli iqna, rom arsebobs e.w.kritikuli temperatura, romlis zeviT metali plastikuria,xolo am temperaturis qvemoT iqceva ise, rogorc myife masala.es dabaltemperaturuli simyife SeniSna s.tipperma 1957 wels. iscdebs ZiriTadad rbil foladze awarmoebda, Tumca analogiaSeiniSna sxva liTonebSic (molibdeni, qromi, tyvia). amave drosSemCneul iqna, rom iseTi liTonebisTvis, rogorebicaa alumini,spilenZi, nikeli, oqro, vercxli, platina da maTi Senadnobebikritikuli temperatura ar arsebobs. isini inarCunebdnenplastikur Tvisebebs nebismier temperaturaze.aseve cnobili gaxda, rom hidrostatikuri wnevamniSvnelovnad moqmedebs liTonebis fizikur Tvisebebze. ammimarTulebiT Catarebuli kvlevebidan unda aRiniSnos Semdegi:T.karmanma 1911 wels hidrostatikuri wnevis qveS SekumSakirqva da marmarilo da aCvena, rom normalur pirobebSi es myifemasalebi maRali hidrostatikuri wnevis zemoqmedebiT plastikurTvisebebs amJRavneben.r.bokerma 1914 wels hidrostatikuri wnevis qveS gamoikvliamarmarilosa da cinkis masalebisagan damzadebuli nimuSebi,romlebic garda zemoTxsenebuli hidrostatikuri wnevisa,imavdroulad ganicdidnen gaWimvisa da grexis deformaciebs. immizniT, rom wyali ar moxvedriliyo zedapirze gaCenil bzarebSi,nimuSebi specialur folgebSi iyo Sexveuli. cdebma aCvena, romwnevis gazrda iwvevda plastikurobis zrdas.p.bridJmenma (1947 – 1952) Caatara mravali eqsperimentiliTonuri da araliTonuri masalebisgan damzadebul nimuSebisgaWimvaze Zalze maRali wnevis pirobebSi (25 000 atm.). am cdebisSedegad man gaakeTa daskvna, rom zogierTi masala kumSvadi iyo,Tumca datvirTvis moxsnis Sedegad TiTqmis aRidgenda pirvandelformas. manve rbili foladisagan damzadebuli nimuSebis141

gamocdisas aRmoaCina, rom hidrostatikuri wnevis zrdisasrRvevis xasiaTi icvleboda.am Temas p.bridJmenma 200-mde statia miuZRvna, sadac aRniSna,rom nimuSebis rRvevisaTvis saWiro zomebis bzarebis warmoqmnasxels uSlis maRali hidrostatikuri wneva, romelic SeiZlebaiTqvas xuravs bzarebs da aferxebs maT ganviTarebas.saWiroa aRiniSnos, rom bevri masala wnevis moxsnis Semdegubrundeba sawyis mdgomareobas, magram ara yovelTvis.zogjer maTSi alotropuli gardaqmnebi xorcieldeba.mravali qimiuri procesebis mimdinareobas maRali wnevebisarseboba uwyobs xels. am sakiTxiT dainteresebuli ariangeologebi da geofizikosebic. r.bredxauerma (1956) SeiswavlamTis qanebis qceva saSualo sididis hidrostatikuri wnevismoqmedebis dros (0 – 15 000 funti/duim 2 ) RerZul kumSvaze, rocaxdeboda myife mdgomareobidan plastikurze gadasvla. esgamokvlevebi saWiro iyo navTobmompovebeli mrewvelobisaTvis,radgan Rrma WaburRilebis burRvis dros adgili aqvs swored amnaSromebSi ganxilul movlenebs.zemoT naxsenebi p. bridJmenis monacemebi TanxmobaSi aRmoCndam.rosi da a.eixingeris (1929) mier Catarebul gamokvlevebTan,romlebic Tujis nimuSebze iqna ganxorcielebuli. analogiurieqsperimentuli Sedegebi miiRes g.hukma (1934) spilenZisa dafoladis nimuSebis grexisas da b.kroslandma (1954) aseve grexiTideformaciebis SeswavliT, romelic spilenZisa da foladisnimuSebis garda, sxva masalebis nimuSebsac iyenebda.am gamokvlevebma daadastura, rom plastikur masalebSiwneva mcire gavlenas axdens denadobis zRvris sidideze daZabvaze. nacrisferi Tujisagan damzadebuli nimuSis grexazegamocdisas, romelic reziniT iyo dafaruli da ganicdidamaRali hidrostatikuri wnevis zemoqmedebas b.kroslandma dav.dirdenma (1958) daaskvnes, rom „mxebi Zabvebi – deformaciebis“diagrama gadis im diagramaze maRla, romelic miiRebaCveulebrivi gamocdebis dros, Tumca mxebi Zabvebi wnevis mkacrad142

proporciulad ar izrdeba. maRali wnevis dros masalaidealurad plastikur mdgomareobas uaxlovdeba.b.kroslandma da a.mitram (1968) daadgines, rom zogierTisaxis foladisaTvis grexis dros, denadobis Zabva, romelicZabva-deformaciis mrudidan ganisazRvreba, umniSvnelod icvlebaTu, imavdroulad nimuSze moqmedebs hidrostatikuri wneva 140 000funti/duim 2 . maT daadastures, rom grexis dros wneva ZiriTadadgavlenas axdens masalis plastikur Tvisebebze. plastikurobaizrdeba daaxloebiT wnevis zrdis proporciulad. maRali wneviszemoqmedebis dros heqsagonaluri kristaluri meseris mqoneliTonebze (mag. magniumi) grexis deformaciebis SemTxvevaSi maTzogierTi araCveulebrivi movlenac SeniSnes, rac gamoyenebuliiqna iseTi procesebis samarTavad, romelTa warmoeba oTaxistemperaturis pirobebSi SeuZlebelia. magaliTad, Tu dawnexasvawarmoebT ara haerSi, aramed maRali wnevis garemoSi, magniumisada vismutisagan damzadebuli nakeTobebi bzarebs ar miiReben.hidrostatikuri wneva, rogorc zemoTac iTqva ewinaaRmdegebabzarebis gaCenasa da ganviTarebas. analogiuri amocanebi ganixilesj. aleqsanderma da b. lenguelma (1964) aluminis nimuSebze.hidrostatikuri wnevis samrewvelo gamoyeneba liTonebisdeformirebisaTvis ganxiluli iqna 1967 wels konferenciaze,romelic maRali wnevebis teqnikur gamoyenebas mieZRvna. agreTvesaerTaSiriso konferenciaze manqanaTmSeneblobis teqnologiebissferoSi, romelic imave 1967 wels gaimarTa. yovelive amisSedegad SeiZleba iTqvas, rom dReisaTvis plastikuri denadobispirobebi kargad aris Seswavlili, rasac ver vityviT rRvevisprocesTan dakavSirebiT. Cven SegviZlia mxolod aRvweroT isfaqtorebi, romlebic ewinaaRmdegebian bzarebis warmoSobas daxels uwyoben denadobis process.miRebulia mosazreba, rom rRveva xdeba mTavari gamWimaviZabvebis an deformaciebis moqmedebis Sedegad, xolo denadobagamowveulia mTavari Zabvebis sxvaobis gavleniT.143

nax. 3. temperaturisa da deformaciis siCqaris gavlenisgrafikuli asaxva tyviis masalisagan damzadebulinimuSis plastikurobaze gaWimvisasnaxazze naCvenebia temperaturisa da deformaciis siCqarisgavlena tyviis masalisgan damzadebuli nimuSis plastikurobazemisi gaWimvis SemTxvevaSi (monacemebi aRebulia a.magnusonisa dav.baldvinis mixedviT).p. ludvigma (1909) SemoiRo naxevrad logariTmulidamokidebuleba gaWimvis dros simtkicis zRvarsa dadeformaciis siCqares Sorisσ = σ +*1σolnε*/εosadac -σ o,σ1da ε * o- mudmivebia.es gantoleba zogad SemTxvevaSi gulisxmobs, rom σ iseswrafad ar izrdeba, rogorc ε * .j.alderma da k.filipsma (1954) Caatares eqsperimentebispilenZze (600 o C), aluminze (500 o C), foladze (930 o C - 1200 o C) daaCvenes, rom adgili aqvs Semdeg damokidebulebasσ = σ .εo* n144

Psadac σo= ; P mimdinare datvirTvis sididea; Xo-XonimuSis sawyisi ganivi kveTis farTobi; n - empiriuli mudmiva.q. makgregorma da j. fiSerma (1946) Semoitanes SesworebasadacTm* *= T (1 − mlnε/ ε ) ,cdis temperaturaTm=dnobis temperaturaaris fardobiTi temperatura;oT - cdis temperatura (temperatura cdis procesSi);*ε -deformaciis siCqare; σo- pirobiTi normaluri Zabva.am gamosaxulebidan Cans, rom deformaciis siCqare izrdebatemperaturis dawevasTan erTad.k. inuiem (1955) gamoiyena gamosaxuleban mσ = σ ε . ε exp( A/Tk), sadac n , m , A da k empiriuli mudmivebia.ol. malverma (1965) warmoadgina gantoleba*ε = σ / E + F[σ − σ ( e)], roca σ > σ (e).man igive gantoleba sxva formiTac mogvawoda* σ σ pε = + D(−1).E σam gantolebebSiσostatikuri denadobis zRvari.oD , F da E p empiriuli mudmivebia, xolo2.3-is daskvnebiTxelkedliani sivrciTi sistemebisaTvis damuSavebuli gamartivebulivariantebi saSualebas iZlevian es gantolebebi miyvaniliqnan wrfiv diferencialur gantolebaTa mimdevrobaze, romelTacvladi koeficientebi Seicaven gadaadgilebas veqtoris mimarT regularulida impulsuri funqciebis saxiT da Sereuli formiT.Txelkedliani sivrciTi kompleqsebis gaangariSebis meTodebigeometriuli da fizikuri arawrfivebis gaTvaliswinebiT saSualebasiZlevian Sefasebul iqnas daZabul-deformirebuli mdgomareobisyvela komponentis cvlileba, kritikuli datvirTvis sidideebi dadatvirTvis procesSi mdgradobis dakargvis forma. amasTan es meTodiufro efeqturia sxva sivrciT an ricxviT-analizur meTodebTan SedarebiT.145

3. დასკვნაCatarebul gamokvlevaTa istoriuli mimoxilvisa daanalizis safuZvelze SeiZleba gavakeToT Semdegi daskvnebi:− samecniero literaturaSi praqtikulad ar aris ganxiluliTxelkedliani sivrciTi sistemebis gaangariSebis meTodebisistoriuli analizi;− zogierTi Sromebis gamoklebiT, ganzogadoebuli impulsurifunqciebi gamoiyenebian mxolod diferencialurigantolebebis CawerisaTvis, magram ara maTi amoxsnebismisaRebad. amave dros wyvetili funqciis SemoReba iZlevagaangariSebis iseTi principulad axali meTodebis miRebissaSualebas, romlebic afarToeben amoxsnadi amocanebis klassda anzogadeben yvela amoxsnad amocanebs erTian meTodologiursafuZvelze. drekadobis wrfivi Teoriis istoriis safuZvelzedReisaTvis kidev ufro meti istoriuli masalis gadmocema damisi analizia gasakeTebeli;− sen-venanisa da almanzis amocanebSi prizmuli RerosTavisufali grexis amocana daiyvaneba harmoniul problemaze,romlis amonaxsnTa meTodebi kargadaa damuSavebuli dagaanalizebuli;− drekadi prizmuli Zelebis daZabul-deformirebulimdgomareobis dadgena, rodesac Zelis boloebze moqmedebsnebismieri ZalTa sistema, aseve warmoadgens drekadobisTeoriis erT-erT ZiriTad da rTul maTematikur amocanas.maTematikuri TvalsazrisiT igi ar aris bolomdegadawyvetili, Tumca e.w. `sen-venanis principis~ daxmarebiTxerxdeba am amocanis gadawyveta, romelic miaxloebiTia da arSeiZleba CaiTvalos zustad. swored, amgvar klasikurmidgomadaa cnobili sen-venanis mosazreba, romelicliteraturaSi damkvidrda `sen-venanis principis~, kerZod kisen-venanis naxevrad Sebrunebuli meTodis saSualebiT.146

daskvnis saxiT xazgasmulia, rom musxeliSvilisgamokvlevebi problemebis farTo klass moicavs, aRniSnulmaSromebma didi gavlena moaxdines meqanikisa da maTematikis rigimimarTulebebis Semdgom ganviTarebaze.n. musxeliSvilis meTodebma drekadobis brtyel TeoriaSigamoyeneba da Semdgomi ganviTareba hpoves s.mixlinis, d.Sermanis,g.savinis, d.vainbergisa da sxva naSromebSi am meTodebissaSualebiT amoxsnilia mravali amocana, romlebmac farTogamoyeneba hpoves teqnikaSi. l.galinis, a.kalandias, i.qarcivaZis,i.Staermanis, r.bancuris da sxva naSromebSi n.musxeliSvilisSedegebma Semdgomi gamoyeneba da ganviTareba hpova sakontaqtoprocesis TeoriaSi.gamokvlevebi Zelebis grexisa da Runvis amocanebSisxvadasxva mimarTulebiT gagrZelebul iqna a. gorgiZis, a. ruxaZisda sxvaTa mier.Txelkedliani sivrciTi sistemebisaTvis damuSavebuligamartivebuli variantebi saSualebas iZlevian es gantolebebimiyvanil iqnan wrfiv diferencialur gantolebaTa mimdevrobaze,romelTa cvladi koeficientebi Seicaven gadaadgilebas veqtorismimarT regularuli da impulsuri funqciebis saxiT da SereuliformiT.Txelkedliani sivrciTi kompleqsebis gaangariSebis meTodebigeometriuli da fizikuri arawrfivobis gaTvaliswinebiTsaSualebas iZlevian Sefasebul iqnas daZabul-deformirebulimdgomareobis yvela komponentis cvlileba, kritikulidatvirTvis sidideebi da datvirTvis procesSi mdgradobisdakargvis forma. amasTan es meTodi ufro efeqturia sxvasivrciT an ricxviT-analizur meTodebTan SedarebiT.147

გამოყენებული ლიტერატურა1. Папкович П.Ф. – Теория упругости. Л-М „Оборонгиз“. 1937, 19392. Крутков Ю. А. – Тензор функции напряжения и общие решения в статикетеории упругости. М. АН СССР. 1949.3. Галеркин Б.Г. – Собрание сочинений в 2-х томах. М. АН СССР. 1952-1953.4. Нейбер Г. – Концентрация напряжений. М-Л. „Гостехиздат“. 1947.5. Аржаных И.С. – Интегральные уравнения основных задач теории поля итеории упругости. Ташкент. АН Уз.ССР. 1954.6. Слободянский М.Г. – Способ приближенного интегрирования уравнений счастными производными и его применение к задачам теории упругости. М.МГУ. 1954.7. Лурье А.И. – Пространственные задачи теории упругости. М. „Наука“.1955.8. Блох В.И. – Теория упругости. Харьков. Харьк. университет. 1958.9. Ляв О. – Математическая теория упругости. Перевод с английского. М-Л.ОНТИ. 1935.10. Гутман С.Г. – К решению задач теории упругости с помощью сферическихфункций. М. „Наука“. 1948.11. Купрадзе В.Д., Гегелия Т.Г., Башалейшвили М.О., Бурчуладзе Г.В. –Трехмерные задачи математической теории упругости и термоупругости.М. „Физматиздат“. 1968.12. Михлин С.Г. – Многомерные сингулярные интегралы и интегральныеуравнения. М. „Физматгиз“. 1962.13. Купрадзе В.Д. – Методы потенциала в теории упругости. М.„Физматиздат“. 1963.14. Лейбензон Л.С. – Собрание трудов. М. Изд. АН СССР. 1951.15. Михлин С.Г. – Численная реализация вариационных методов. М. „Наука“.1966.16. Ростовцев Н.А. – К задаче о кручении упругого полупространства. М.„Прикладная математика и механика“. 1955.17. Грилицкий В.В. – Кручение двухслойной упругой среды. Киев.„Прикладна механика“. VII . 1961.148

18. Виноградов А.И. – К лекциям по устойчивости прямолинейных стержней.Харьков. Инст.ж. д. тр. 1952.19. Агарев В.А. – Метод начальных функций для двумерных краевых задачтеории упругости. Киев. АН УССР. 1963.20. Прокопов В.К. – Осесиметриальная задача теории упругости избранногоцилиндра. М. „Наука“.1950.21. Соляник-Красса К.В. – Введение в механику деформируемого твердоготела. Л. Ленуниверситет. 1960.22. Тренин С.И. – Построение метода решения ряда осесимметричных задачтеории упругости. М. Вестник МГУ. 1952.23. Детинко Ф.М. – Прочность и колебания Эл. машин. Л. „Энергия“. 1953.24. Космодамианский А. С. – Некоторые задачи теории упругости иконцентрации напряжений. Киев. „Наукова думка“. 1962.25. Уздалев А. И. – Некоторые задачи термоупругости анизотропного тела.Саратов. Саратуниверситет. 1962.26. Александров А.Я., Соловьев Ю.И. – Пространственные задачи теорииупругости. М. „Наука“. 1962.27. Соляник-Красса К.В. – Осесиметричная задача теории упругости. М.„Стройиздат“. 1962.28. Динник А.Н. – Избранные труды. Киев. АН УССР. 1952, 1955, 1956.29. Кауфман Р.Н. – Решение некоторых краевых задач математической физикидля слоя с шаровыми полостями. Минск. БГУ. 1958.30. Александров А.Я. – Пространственные задачи теории упругости. М.„Наука“. 1962.31. Шапиро С.Г. – Упругое равновесие параболида вращения. М. „Прикладнаяматематика и механика“. 1950.32. Форсман Н.А. – О концентрации напряжений в растянутом стержнекруглого сечения. М. АН СССР. 1958.33. Ворович И.И. – Некоторые математические вопросы теории пластин иоболочек. М. „Наука“. 1966.34. Гавра Д.Л. – Некоторые случаи кручения призм с криволинейнымиконтурами. Л. Лениндустр. инст. 1939.35. Скоробогатько А.А. – О кручении цилиндрических валов с круговымивыточками. М. АН СССР. 1958.149

36. Угодчиков А.Г. – Кручение полных призматических стержней. М.Прикладная механика. 1956.37. Шерман Д.И. – Об одной задаче кручения. Доклады АН СССР №5. 1948.38. Шерман Д.И. – К вопросу о кручении эллиптического бруса, продольноослабленного эллиптической же полостью. М. Инж. сборник 25. 1959.39. Степанов Р.Д., Шерман Д.И. – Кручение круглого бруса, ослабленногодвумя цилиндрическими круговыми полостями. М. Инж. сборник 11. 1952.40. Амензаде Ю.А. – Кручение и изгиб призматических брусьев с сечениями,представляющими собой двусвязные области некоторого вида. Тб. АНГССР. 1958.41. Капанян Л.К. – О кручении некоторых полых призматических стержней.Ереван. Изв. АН Арм. ССР. 1952.42. Уфлянд Я.С. – Биполярные координаты в теории упругости. М-Л.„Гостехиздат“. 1950.43. Леонов М.Я. – К теории чистого кручения. ВМПМ. 1960.44. Китовер К.А. – Новая форма решения задачи о кручении стержня. Л.Ленполитех. инст. 1954.45. Панов Д.Ю. – О кручении стержней, поперечное сечение которыхограничено двумя коническими сечениями. М. Труды ЦАГИ. 1936.46. Слободянский М.Г. – Способ приближенного интегрирования уравнений счастными производными и его применение к задачам теории упругости.ПММ. 1939.47. Слободянский М.Г. – пространственные задачи теории упругости дляпризматических тел. М. МГУ. 1940.48. Слободянский М.Г. – Определение производных искомых функций прирешении задач методом конечных разностей. ПММ XV. 1951.49. Пивоваров А.М. – Концентрация касательных напряжений при кручениипризматических стержней. М. Прикладная математика и механика. 1953.50. Ветчинкин В.П. – Избранные труды. М. Изв. АН СССР. 1956.51. Лейбензон Л.С. – Собрание трудов. М. АН СССР. 1951.52. Канторович Л.В. – Один прямой метод приближенного решения задачи оминимуме двоиного интеграла. М. Изв. АН СССР. 1933.53. Чепова Т.К. – Приближенное решение задачи кручения некоторыхпризматических стержней. ПММ 1. 1937.150

54. Лурье А.И. – Труды Ленинградского индустриального института. №3.1939.55. Гулканян Н.О. – О кручении призматических стержней с прямоугольнымсечением при наличии продольных трещин. Ереван. Изв. АН Арм. ССР.1953.56. Галин Л.А. – Упругопластические задачи. М. „Гостехиздат “. 1939.57. Бондаренко Б.А. – Задачи кручения стержней. М. Инст. механики. 1956.58. Галимханов К.Г. – Новый метод расчета лысочных валов на кручение. Уфа.Авиац. инст. 1955.59. Амен-заде Ю.А., Саркисов Г.М. – Приближенное исследование кручениятрубы. Баку. Труды Азер. гос. пед. инст. 195260. Геонджян Г.П. – К теории стесненного кручения сплошныхпризматических стержней. Ер. АН Арм. ССР. 1959.61. Авазашвили Д.З. – О применении теории функций комплексногопеременного к задачам кручения и изгиба. М. ПМН. 1940.62. Шерман Д.И. – Изгиб поперечной силой эллиптического бруса,ослабленного продольно-круговой цилиндрической полостью. М. Инст.механики АН СССР. 1953.63. Амензаде Ю.А. – Местные напряжения при кручении круглогопризматического бруса с эллиптическим несоосным отверстием. ДАНСССР 119, №6. 1958.64. Арутюнян Н.Х. – Решение задачи о кручении стержней полигональногопоперечного сечения. ДАН Арм. ССР 9, №2. 1958.65. Арутюнян Н.Х., Абрамян Б.Л. – Кручение упругих тел. М. Изд. Физматлит.1963.66. Гулканян Н.О. – О кручении призматических стержней прямоугольногосечения с несимметричным прямоугольным вырезом. Ер. АН Арм. ССР.1957.67. Векуа И.Н., Рухадзе А. К. – Задача кручения кругового цилиндра,армированного продольным круговым стержнем. М. Изв. АН. СССР №3.1933.68. Рухадзе А.К. – Кручение и изгиб бруса, составленного из двух упругихматериалов, разграниченных эпитрохоидой. Тб. ТГУ. 1935.69. Кутателадзе Г.А. – Кручение и изгиб поперечной силой цилиндрическогобруса, составленного из различных упругих материалов, поперечноесечение которого разграничено гипотрохоидами. Тб. Труды ГПИ. 1956.151

70. Чобанян К.С. – Кручение составного вала переменного диаметра. ДАН АР.ССР №3. 1958.71. Сухаревский И.В. – К задаче о кручении составного многосвязного бруса.М. Инст. механики АН СССР. 1954.72. Лехницкий С.Г. – Теория упругости анизотропного тела. М. Гостехиздат.1950.73. Арутюнян Н.Х. – Приближенное решение некоторых задач о кручениианизотропных стержней. Сообщ. Инст. матем. и мех. АН Арм. ССР. 1948.74. Минасян Р.С. – О кручении и изгибе анизотропных призматическихстержней с поперечным сечением в виде параллелограма. Ер. АН Арм.ССР. 1958.75. Галфаян П.О., Чобанян К.С. – Кручение полого прямоугольного стержня стонким усиливающим покрытием. Ер. АН Арм. ССР. 1959.76. Новожилов В.В. – Теория упругости. М. „Наука“. 1957.77. Соляник-Красса К.В. – Кручение валов переменного сечения. М.Физматиздат. 1949.78. Положий Г.Н. – Вариационные теоремы краевых задач теории кручениявалов переменного сечения. М. „Наука“. 1957.79. Зволинский Н.В., Риз Л.П. – Кручение цилиндрического стержня силами,распределенными по его боковой поверхности. М. Изв. АН СССР. 1939.80. Костандян Б.А. – О кручении полого ступенчатого вала. Ер. Изв.АН Арм.ССР. №3. 1956.81. Горгидзе А.Я. – Некоторые обобщения задач кручения и изгиба составныхбрусьев. Тб. Мецниереба. 1955.82. Мецугов В.Х. – К задаче кручения растянутого призматического бруса,составленного из различных упругих материалов. Тб. Труды ГПИ. 1954.83. Джанелидзе Г.Ю. – Принцип Сен-Венана. Труды ЛПИ им. Калинина.№192. 1958.84. Хатиашвили Г.М. – Об изгибе моментами и растяжения силой составныхбрусьев с прямолинейной анизотропией. Тб. Сообщения АН ГССР. 1965.85. Хатиашвили Г.М. – О задачах Сен-венана для однородных анизотропныхбрусьев. Тб. Сообщения АН ГССР. 1963.86. Хатиашвили Г.М. – Задачи Альманзи-Мичелла для однородных исоставных. Тб. Мецниереба. 1983.152

87. Тимошенко С.П. – Использование новых методов при исследованииустойчивости конструкций некоторых мостов. М. „Наука“. 1910.88. Попов Г.Я., Ростовцев Н.А. – Контактные (смешанные) задачи теорииупругости. Киев. Мат. журнал. 1966.89. Леонов М.Д. – Основы механики упругого тела. Фрунзе. АН Киргиз.ССР.1963. (Монография).90. Штаерман И.Я. – Контактная задача теории упругости. М-Л. Гостехиздат.1949. (Монография).91. Галин Л.А. – Контактные задачи теории упругости. М. Гостехиздат. 1953.(Монография).92. Уфлянд Я.С. – Интегральные преобразования в задачах теории упругости.Л. „Наука“. 1967.93. Рвачев В.Л. – О давлении на упругое полупространство штампа, имеющегов плане форму клина. М. Прикладная математика и механика. 1949.94. Моссаковский В.И., Губенко В.С. – Давление осесиметричного кольцевогоштампа на упругое полупространство. М. „Прикладная математика имеханика“. 1960.95. Коренев Б.Г. – Вопросы расчета балок и плит на упругом основании. М.Госизд. лит. по стр.-ву. 1954.96. Александров В.М., Сметанин Б.И. – Тонкие концентраторы напряжений вупругих телах. М. „Наука“. 1965.97. Савин Г.Н. – Механика деформируемых тел. Избранные труды. Киев.„Наукова думка“. 1979.98. Лебедев Н.Н., Уфлянд Я.С. – Осесимметричная контактная задача дляупругого слоя. М. „Прикладная математика и механика“. 1958.99. Девнорович В.И. – Пространственные контактные задачи теорииупругости. Минск. Белгосуниверс. 1964.100. Грилицкий Д.В., Кизыма Я.М. – Осесимметричные контактные задачитеории упругости и термоупругости. Львов. Высшая школа. 1964.101. Александров В.М. – Контактные задачи теории упругости. М. „Наука“.1967.102. Панасюк В.В. – Предельное равновесие хрупких тел с трещинами. Киев.„Наукова думка“. 1986.153

103. Михайлов Б.К., Кипиани Г.О. – Устойчивость трехслойных пластин свырезами. М. Строит. механика и расчет сооружений. №4. 1989.104. Попов Г.Я. – Математические проблемы контактных задач. Одесса. Вышашкола. 1964.105. Мусхелишвили Н.И. Некоторые основные задачи математической теорииупругости. М. „Наука“. 1966.106. Космодамианский А. С. – Очерки по истории механики. М. Прикладнаяматематика и механика.. 1964.107. Моисеев Н.Д. – Очерки развития механики. Изв. Моск. Университета.1961.108. Гольденвейзер А.Л. – Свободные колебания тонких упругих оболочек. М.„Наука“. 1979.109. Савин Г.Н., Флейшман Н.П. – Пластинки и оболочки с ребрами жесткости.Киев. „Наукова думка“. 1964.110. arabiZe m., daTuaSvili a. – fermebis (wamweebi) gaangariSebisistoriis sakiTxisaTvis (XIX, XX saukuneebis mijnaze). Jurnali„saistorio vertikalebi“, 1 (10), 2006.111. bacikaZe T., daTuaSvili a., arabiZe m. – maRali Zabvis eleqtrogadamcemixazebis sayrdenebis qarsawinaaRmdego mdgradobissakiTxisaTvis. Jurnali „keramika“, 1 (15), 2006.112. bacikaZe T., kakuSaZe a. – praqtikuli meTodebis damuSavebaanizotropuli koWebis, filebisa da garsebisgasaangariSeblad. spi. Tb., 1975.113. gorgiZe a., ruxaZe a. – dagrexili Zelis wyvilZaliT Runvisamocana. Tb., moambe. 1944114. daTuaSvili a., niJaraZe j. – drekadobis Teoriis brtyeliamocanis dasma da misi konformuli asaxviT amoxsnisistoria. Jurnali „mecniereba da teqnologiebi“, # 4-6, 2006115. vekua i. – akademikosi nikoloz musxeliSvili. Tb. 1991.116. Векуа И.Н. – Теория тонких пологих оболочек переменной толщины. Тб.1965.117. Горгидзе А.Я. – Метод последовательных приближений к плоской задачетеории упругости. Доклады АН СССР. 1934.118. Байда Э.Н. – Общие решения теории упругости и задачи напараллелепипеде и цилиндре. Л. Лен. унив. 1961.154

119. Динник А.Н. – Устойчивость упругих систем. Л-М ОНТИ 1935.120. Девнорович – В.И. Пространственные контактные задачи теорииупругости.121. Лейбензон Л.С. – Курс теории упругости. М. „Гостехиздат“. 1947.122. Лейбензон Л.С. – Вариационные методы решения задач теории упругости.М., 1943.123. Лехницкий С.Г. – Анизотропные пластинки. М. „Гостехиздат“. 1947.124. Лехницкий С.Г. – Кручение анизотропных и неоднородных стержней. М.Наука, 1971.125. Локшин А.З. – Изгиб и устойчивость пластин и круговых цилиндрическихоболочек. Л. ОНТИ. 1955.126. Лурье А.И. – Пространственные задачи теории упругости. М. 1955.127. Лурье А.И. – Теория упругости. М. „ Наука“. 1970.128. Лурье А.И. – Статика тонкостенных упругих оболочек. М-Л. Гостехиздат.1947.129. Механика деформирующих сред. Межвузовский научный сборник. Вып. 5, 1978.130. Михлин С.Г. – О распределении напряжений в полуплоскости сэллиптическим вырезом. Л. Ленполитех. инст. 1934.131. Михлин С.Г. – Некоторые новые вопросы механики сплошной среды. Л.„Наука“. 1938.132. Михлин С.Г. – Приложения интегральных уравненый к некоторымпроблемам механики, математической физики и техники. М. „Наука“.1947.133. Мурье А.М. – Пространственные задачи теории упругости. М.„Физматгиз“. 1955.134. Мурье А.М. – Статика тонкостенных и упругих оболочек. М. „Наука“.1947.135. Панасюк В.В. – Взаимодействие жестких линейных включений и трещин вдеформируемом теле. Киев. „Наукова думка“. 1983.136. Панасюк В.В. – Распределение напряжений около трещин в пластинках иоболочках. Киев. „Наукова думка“. 1976.137. Полозков А.А. – Изгиб, устойчивость и колебания подкрепленных пластин.Ростов. 1971.155

138. Панасюк В.В. – Предельное равновесие хрупких тел с трещинами.„Наукова думка“. 1968.139. Прокопов В.К. – Осиметриальная задача теории упругости изобранногоцилиндра. Л. 1950.140. Работнов Ю.Н. – Механика деформируемого твердого тела. М. „Наука“.1961.141. Рвачев В.Л., Синекоп Н.С. – Метод R-функций в задачах теории упругостии пластичности. Киев. „Наукова думка“. 1968.142. Савин Г.И. – Концентрация напряжений около отверстий. М-Л.Гостехиздат, 1951.143. Саркисян В.С. – Некоторые задачи теории упругости анизотронного тела.Ереван. Университет. 1970.144. Сен-Венан Б. – Мемуар о кручении и изгибе призм. М. Гостехиздат.1959.145. Соляник-Красса К.В. – Осесимметричная задача теории упругости. М. Изд.АН СССР. 1987.146. Тимошенко С.П. – История науки о сопротивлении материалов. ГИЗТТА.М. 1957.147. Тимошенко С.П. – Пластинки и оболочки. М-Л. Гостехиздат. 1948.148. Тимошенко С.П. – Прикладная теория упругости. Л. Гостехиздат. 1930.149. Тимошенко С.П. – Теория упругости. М-Л. 1934, 1975, 1979.150. Тимошенко С. П. – Устойчивость стержней, пластин и оболочек. М.Гостехиздат. 1971.151. Тимошенко С.П. – Вопросы устойчивости упругих систем. Л. Гостехиздат.1935.152. Филоненко-Бородич М.М. – Теория упругости. М.-Л. Гостехиздат. 1947.153. Филоненко-Бородич М.М. – Сопротивление материалов. М. Физматиздат.1949.154. Ширяев Е.А. – О кручении круглого бруса с трещиной на дуге окружностиили по радиусу. ПММ 20. №4. 1956.155. Статические и динамические задачи теории упругости и пластичности.Баку. Сборник статьей. 1968.156. Трещиностойкость материалов и элементов конструкций. Киев. „Науковадумка“. 1980. материалы симпозиума.157. Некоторые вопросы механики деформируемых сред. М. Изд. АН СССР. 1959.156

158. Упругость и пластичность. М. Инст. тех.инж. 1967.159. Прочность, устойчивость, колебания. М. „Машиностроение“. 1968. 3 тома.160. Некоторые задачи теории упругости. Тб. ТГУ. 1975.161. Развитие теории контактных задач в СССР. М. „Наука“. 1976.162. Некоторые задачи теории упругости о концентрации напряжений идеформаций упругих тел. Саратов Изд. Универ. 1970.163. Прикладные вопросы теории упругости и вязко-упругости. Пермь.Политехн. инст. 1971.164. Методы расчета оболочек. Киев. „Наукова думка“. 1980.165. Пространственные задачи теории упругости и пластичности. 6 томм. Киев.„Наукова думка“. 1984.166. Теория оболочек с учетом поперечного сдвига. Казань. КГУ. 1977167. Механика деформируемого тела. М. „Наука“. 1986.168. Механика твердых деформируемых тел. М. ВИНИТИ. 1973.169. Исследования по упругости и пластичности. Сборник 3. Л. 1964.170. Исследования по упругости и пластичности. Сборник 12. Л. 1978.171. Исследования по упругости и пластичности. Сборник 13. Л. 1980.172. Исследования по упругости и пластичности. Сборник 15. Л. 1986.173. Исследования по упругости и пластичности. Сборник 16. Л. 1990.174. Мойзель В.М. – Температурная задача теории упругости. Киев. АН УССР.1951.175. Маслов Г.С. – Расчеты колебаний валов. М. „Машиностроение“. 1968.176. Шерман Д.М. – Об одной особой задаче теории потенциала. М. ДокладыАН СССР. 1954.177. Карцивадзе И.Н. – Основные задачи теории упругости для упругого тела.М. Гостехиздат. 1943.178. Араманович И.Г., Лунц Г.А., Эльсгольц Л.Э. – Функция комплексногопеременного. Операционное исчисление. Теория устойчивости. М.„Наука“. 1965.179. Бабаков И.М. – Теория колебаний. М. „Наука“. 1965.180. Гузь А.Н. – Пространственные задачи теории упругости и пластичности. В6 томах. Киев. Изд. АН. 1985181. Гузь А.Н. – Методы расчета оболочек. В 5 томах. М. „Наука“. 1982.157

182. Огибалов Л.М. – Изгиб, устойчивость и колебания пластинок. М. МГУ.1958.183. Огибалов Л.М. –Пластинки и оболочки. М. МГУ. 1969.184. Каландия Л.И. – Математические методы двумерной упругости. М.„Наука“. 1973.185. Коши О.А. – Алгебраический анализ. Лайпциг. 1864.186. Купрадзе В.Д. – Трехмерные задачи теории упругости. Тб. ТГУ. 1968.187. Михлин С.Г. – Плоская задачи теории упругости. М-Л АНСССР. 1935.188. Угодчиков Ф.Г. – Кручение полых призматических стержней. М.Прикладная механика. 1956.189. Угодчиков Ф.Г. – Решение краевых задач теории упругости методамианалитических функций. Горький. Гос. унив. 1969.190. Горгидзе А.Я., Рухадзе А.К. – Об одном численном решении интегральныхуравнений плоской задачи теории упругости. Тб. Сообщения АН. 1940.191. Космодамианский А.С. – Некоторые задачи теории упругости оконцентрации напряжений. Автореферат. д.т.н. Киев. АН УССР. 1963.192. Вайнберг Д.В. – Механические колебания. Киев. АН УССР. 1953.193. Вайнберг Д.В. – Расчет оболочек. Киев. Госстройиздат. 1961.194. Вайнберг Д.В. – Расчет пластин. Киев. Кудивельник. 1970.195. Каландия А.И., Жгенти В.С. – О плоских задачах моментной теорииупругости. Тб. Инст. математики. 1967.196. Рвачев В.Л. – К задаче о давлении на упругое полупространство штампа сплоским основанием. М. „Прикладная математика и механика“. 1957.197. Тренин С.М. – О решениях уравнений равновесия осесимметричной задачитеории упругости. М. Вестник. МГУ. 1953.198. Зволинский К.В., Шапиро Г.С. – Динамика пластичных сред. М. „Наука“.1963.199. Динамические задачи теории упругости. Л. Лен. унив. 1953.200. Материалы конференции по математической теории упругости в Тбилиси.Тб. фил. АН СССР. 1940.158