Ha npasax pyKonncH

BYPMAIUEBA HATAJ1B5I BJIA^HMMPOBHA

AHAJIH3 HAHP5I>KEHHOrO COCTOSHHS

H HPEAEJIbHblX HAFPySOK CTEP^CHEBOH CHCTEMbI

C 3JIEMEHTOM H3 PA3ynPO^H5IIOIIi;ErOC5I

MATEPHAJIA HPH TPEXOCHOM PACTH>KEHHH

Cnen,HajibHocTb 01.02.04 -- MexaHHKa,n,e(popMHpyeMoro

ABTOPEOEPAT

flHCcepTau,HH na concKaHHe yneHOH CTenenn nayK 2013 Pa6oTa BbinojiHena B Oe^epajibHOM rocya,apCTBeHHOM ManiHHOBeflenHH YpajibCKoro OTfleneHHH POCCHHCKOH aKa^eiviHH nayK Hayqnbin pyKOBOflHTejib: CTpy^KanoB Bajiepnft,fl,OKTOp CpH3HKO npocpeccop OcpnnHajibHbie onnoneHTbi: SajiasHHCKnft AjieKcan^p FeoprneBiiq ^,OKTOP TexHHqecKnx nayK, npocpeccop, OFBYH HncTHTyT MannraoBefleHHH YpajibCKoro OTflejiennH POCCHHCKOH nayK, aaBe-flyioinHH jia6opaTOpH6H CpHSHKO-MaTeMaTHHeCKHX npocpeccop, OFBOY BFIO "CaMapcKHii H HHCpOpMaTHKH opraHH3aH,H5i: OFAOY BI1O "YpajibCKHH yHHBepcHTeT HM. nepBoro Hpesn^eHTa POCCHH B.H. Ejibu,Hna " 3aiu,HTa ^HCcepTaii,HH COCTOHTCM 10 ^eKa6pa 2013 ro^a B 15 qacoB 30 MHHyT na 3ace^,aHHHflHCcepTau,HOHHorocoseTa /], 004.023.01 npn Oe^epajibHOM rocy^apCTBCHHOM Gio^xeTHOM yqpe>K,ii,eHHH nayKH HncTHTyTe MamnnoBeflenHH YpajibCKoro OT^ejienHH PAH no a^pecy: 620049, r. EKaTepnnGypr, yji. KOMCOC flHCcepTau,HeH MOJKHO osnaKOMHTbCH B SnGjiHOTeKe Oe/],epajibHoro rocyflapCTBeHHoro 6io,ii,>KeTHoro yqpejKfleHHH nayKH HncTHTyTa YpajibCKoro OTflejienHH POCCHHCKOH aKa,n,eMHH nayK.

AsTOpecpepaT pasocjian " _ " HOH6pH 2013 ro^a.

YqenbiH ceKpeTapb /pccepTauHOHHoro TexHHqecKHx nayK, npocpeccop ^to K KonoBajiOB A.B.

O6ni,aa xapaKTepncTHKa pa6oxbi T6MBI HCCJie/i,OBaHHH. IlocTOHHHoe y>K6CToqeHHe Tpe6oK KaqecTBy sjieMCHTOB KOHCTpyKHHH, npe^nojiaraeT BHe,n,peHHe B npaKTHKy npoeKTHpOBaHHa see 6ojiee coBepmenHbix MCTO^OB pacqe'Ta HX npoqHOCTH, flOJiroBeqnocTH, Hafle^KHocTH H jKHByqecra. CosflanHe HOBbix MCTO^OB 6a3HpyeTCH Ha BBe^enHH B paccMOTpenne CBOHCTB MaTepnajiOB, ne yqHTblBaCMblX B TpaflHHHOHHblX TeOpHHX MexaHHKH ^,6(pOpMHpyeMOrO TBCpflOro Tejia, nanpHMep, CBOHCTBa MaTepnajiOB, npOHBji5nom,HecH B HajinqHH HHcnayqacTKOB na ^narpaMMe ^,ecJ3opMHpOBaHH5i — CTa/pin pasynpoqHeHHH.

Heo6xoflHMOCTb nccjie^OBaHHa saKpHTHqecKoro HOBe^eHHH CHCTeM o6ocnoT6M, qTO SHanne CBOHCTB TCJI na saKpHTHqecKoft CTa,n,HH RedpopuunosBOJiHCT nojiHee HcnoJIbsoBaTb HMeK)LU,Heca npoqHOCTHbie pesepBbi, npHBO^HT B HTore K noBbiuieHHio 6esonacHOCTH MexannqecKHX CHCTCM, BKjnoqaioiHHx Tejia, ^Jia KOTopbix ocyui,ecTBHMO saxpHTHqecKoe ^,e(popMHpoBaHne. IIpH 3TOM CTa^HH paaynpoqHeHHH xapaKTepnsyeTca HaKonjieHHeM B MaTepnajie MHKpOTpeui;HH H BOSHHKHOBeHneM MaracTpajibHOH Tpeiu,HHbi, KOTOpaa B pesyjibTaTe HBJiaeTCH OCHOBHOH npHqHHOH CHH>KeHH5i narpysKH.

B 3KcnepHMeHTajibHbix H TeopeTHqecKHx pa6oTax POCCHHCKHX H sapy6e>KHbix yqenbix, cpe,n,H KOTopbix C.^l,. BOJIKOB, H.C. BoponiOK, B.A. HGparHMOB.

K).B. Ka^auieEHq,,11,. B. KJHOHIHHKOB, B.B. HOBO^HJIOB, A. A. JIe6epB, H.F.

HaycoB, Jl.B. HHKHTHH, E.H. Pbi»caK, B.B. CTpy>KanoB, B.IT. Pa^qeHKO, B.3.

BHJibfleMan, Z.P. Bazant. J.Bobinski, M. Brocca, D.C. Drucker, R.H. Evans, E.

Smith, R.Y. Xiao H ^pyrnx 6biJia ycTaHOBjiena npHHUHnnajibHafl BO3MO>KHOCTb SKcnepHMeHTajibHoro nocTpoennH flnarpaiviMbi ^ecpopMnpOBanHH c na^aiomefi f(O HyjI5I BCTBbK), H yCTaHOBJICH, HO KpaHH6H M6p6, Ha KaqeCTB6HHOM ypOBHC 3(f)(peKT OT BKJiioqeHHfl B paccMOTpenne saKpHTHqecKOH C BaHHM (pasynpoqneHHH), 3aKjnoqaiou],HHC5i B yToqneHHH snaqenHH Hecyuj,eH cnoco6nocTH H Hanpa^KenHO-^ecpopMHpyeMoro COCTOHHHH, npe^uieCTBytoinero paspymeHHio.

SaKpHTHqecKoe fle(popMHpOBaHHe saBe^OMO HeycTonqHBO, HO Bbie COCTO5IHH5I MaTCpnajia MoryT 6biTb peajiHSOBanbi, CCJIH STOT B cocTase ycToftqHBOH MexannqecKOH CHCTeMbi.

COCTOSIHHH cym,ecTBeHHO CBHsana c Heo^HOMepHOCTbio Teji H ne HM66T OflHOMepHblX aHajIOFOB.

Hcnojib3OBaHHe saxpHTHqecKHx xapaKTepncTHK CBHsano ne TOJibKO c Tpy^,HOCTHMH 9KcnepHMeHTajibHoro xapaKTepa, HO H c MaTeMaTHqecKHMH npo6jieMaMH, ne xapaKTepnbiMH fljia Tpa^,Hu,HOHHOH MexaHHKH ^ecpopMHpyeMoro TBep^oro Tejia. BTO B OCHOBHOM nee^HHCTBeHHOCTb H HeycTofiqHBOCTb peuieHHii HCJIHHeftHbix ypaBHeHHH paBHOsecHH. TaKHe sa^aqn ne y^OBjieTBOpaiOT ycjiOBHHM KOppeKTHOCTH AflaMapa. HosTOMy nocTyjiaT ^pyKKepa, BbinojineHHe KOTOPOro rapaHTupyeT KoppeKTHOCTb sa^aq, HBJIJIJICJI, fla H HBJi5ieTC5i, ycjiOBHCM OT6paKOBKH MOflejieft. OflHaxo, KaK noKasano B neKOiopbix pa6oTax CTp B.B., BHJib^eMana B.3. H flp., HeBbinojmeHHe nocxyjiaTa ^pyKKepa ne peiueHHio OT^ejibHbix sa^an MexaHHKH. KpoMe TOFO, noKasano, TITO yqeT pasynpoqHeHHH MO^KCT nosBOJIHTb pacqHTbmaTb npe^ejibHyio narpysKy, 6jiH3Kyio K peajibHOCTH. /JajibHeHiuee pacnpocTpaHHne Bbi^BHHyTbix npe,n,nojio^KeHHH Tpe6yeT peuienHH KOHKpeTHbix neo^HOMepHbix sa^an, na KOTopbix BOSMOKHO 6biJio 6bi BbMCHHTb see ocoGeHHocTH TeopHH pa3ynpoqH5HOui,eroca Tejia. TaKne npHMepbi narjiH^HO 6bifleMOHCTpnpOBajiHnocTaHOBKH sa^an, noflXOflbl H MCTOflbl HX pemeHHH H HO3BOJI5IJIH 6bl HCCJI6flOBaTb 9(p(peKTbI, CKpblTbie npn o6meM paccMOTpennH. KpoMe Toro, nocTpoennbie MCTO^bi H ajiropnTMbi y>Ke na flannoM 3Tane HCCJie^OBaHHa npo6jieM pasynpOHHenHH MaTepnajia B 9JieMeHTax KOHCTpyKii,HH MoryT 6biTb BKJiiOHeHbi B npaKTHKy npoeKTHpofiaHHa OT,fl,eJIbHbIX KOHCTpyKU,HH.

I^ejibK) /i,aHHOH ^nccepTai^HOHHOH paSoTbi HBJIH6TCH paspa6oTKa MeTO^OB anajiHsa nanpH>KeHHoro cocTOflHHfl H pacqexa npe^ejibHbix narpysoK rpa^HeHTHOH flHCKpeTHoft MexaHHHecKoii CHCTCMbi, MOflejinpyromefi: ^ecpopMaii,HOHHoe HOBefleHne H paspymeHne npn TpexocnoM pacTa^KeHHH B ynpyron cpefle sjieMeHTa HS pa3ynpoqH5iioui,eroc5i Maxepnajia (sjieMenTbi Tpy6 H c(J)epHqecKHx cocy^OB 6ojibiuoro flHaMeTpa npn HHH), H o6lHHX npHHU,HHOB HO3BOJIHIOHJ,HX y^KG Ha ^aHHOH CTa^HH BHCflpHTb B npaKTHKy npoeKTHpOBanHa BTH MGTOflHKH fljia pemenHfl anajiorHHHbix 1. OnncaTb ynpoqHeHHe H pasynponneHHe MaTepnajia npn ero TpexocnoM fle4)opMHpOBaHHH eflHHbiM BbinyKjio-BornyTbiM HOTeHii,HajiOM, no3BOjiHiouj,HM na o6enx CTa^nax sanncaTb CB5i3b Me^K^y HanpH>KeHH5iMH H ^ecpopMaujHHMH B KOHGHHblX COOTHOHieHHH.

2. PasBHTb 34)(J)eKTHBHbie qncjieHHbie MCTO^bi pacneTa napaMeTpOB scex paBHOsecHH, B TOM qncjie H HeycTOHHHBbix, B npHMenenHH K paceCKOH CHCT6M6.

3. Paspa6oTaTb MeTO^HKy, no3BOJi5nouj,yio naxo^HTb npe^ejibHbie snaqeHHH narpysoK, 6jiH3KHe K peajibHOCTH, 6es peuieHHJi CHCTOM HejiHHenHbix ypaBneHHH paBHOB6CH5I.

Haynnaa HOBHBna saKjuonaeTca B cjie/jyiomeM:

1. HoKaaaHa BOSMO^KHOCTB nocTpoennfl BbinyKJio-BornyToro noTeHii,Hajia, KOTOpbift c e^HHbix no3Hii,HH QHHCbiBaeT CBOHCTBa MaTepHajia npn ero ynpoqH6HHH H paaynpOHHeHHH B pesyjibTaTe aKTHBnoro Tpexocnoro pacTH>KeHH5i.

2. YcTaHOBJieHO, HTO ^anHbiH noTenuHaji onpe^ejiaeT KOHe^nyio MOCTb M6>Kfly ^eCJDOpMai^HMMH HpH aKTHBHOM Harpy>K6HHH H KOTOpyio MO>KHO TpaKTOBaTb KaK ^Hcj^epeHHHpyeMoe OTo6paJK6HHe npocTpanCTBa flecpopMaijHH B npocTpancTBO nanpfljKeHHH, o6jia^,aion],ee oco6eHHOCT5iMH.

BTH oco6eHHOCTH CBflsaHbi c BbipOKflCHHeM MaTpHijbi 5lKo6n noTGHijHajia, npnH6M TOHKH BblpCtfKfleHHOCTH COOTB6TCTByK)T IIOrpaHHHHOMy COCTO5IHHK) MaT6pnajia (nepexofl OT ynponneHHa K pasynpOHHGHHio).

3. BbinHcana noTemjuajibHafl cpyHKipiH fljia Bceft MexaHHTiecKoft H ycTanoBJieHO, HTO nopo^K^aeMbie eio ypaBH6HH5i paBHOBecua HMCIOT KO pemeHHH. IlpHBe^eHa MeTOflHKa onpeflejiemM nucjia peiueHHH paBHOB6CM5i) fljia sa^aHHOH BHeuiHeft narpysKH.

TeopeTunecKaa sna^HMOCTB HCCJie^OBaHHii o6ocnoBaHa TGM, npe^,Jio»c:eHHbie MeTOflbi H no^xoflbi MoryT 6biTb HcnojibSOBanbi ^JIH ^,ajibHefluiero pasBHTHH TeopeTHHecKHX nojio*eHHH MexaHHKH pasynpOHHaiomHxca MaH paspaGoTKH 9(p(peKTHBHbix MeTO^OB pacneTa na npo^HOCTb H >KHpaajiHHHbix KOHCTpyKi^ufi, KOTOpwe BCJie^CTBHe yneTa CTa^HH pasynosBOJiflT nojiHOCTbio ncnojibsoBaTb pecypc MaTepnajia.

PaapaGoTana HHCJiennaH npoi],eflypa Bbi6opa Heo6xo^HMoro Hbix npn6jiH>KeHHii ^JIH peajiH3au,HH HTepan,HOHHOH cxeMbi K aa^ane 06 onpe^ejieHim napaMexpOB scex paBHOBecHbix COCTOHCCJieflyeMOH MexaHHHecKoii cucTeMbi.

PaspaGoTana MeTOflHKa pac^eTa npe^,ejibHbix narpysoK, nosBOJiHiomaa H3e»caTb peineHHJi CHCTCM HejiHHefiHbix ypaBHGHHH paBHOBecna. MeTOflHKa OCHO Bana na ncnojibsoBanHH cenapaTpucbi noTenijHajibHOH 4)yHKu,HH CHCT6MbI.

H3jio>KeHHbie peayjibTaxbi BHOCST Heo6xoflHMyio HcnocTb o u,eji5ix H flajibHefiinero Hcnojib3OBaHH5i CTa^HH pasynpOMHeHH5i npn npaKTH^ecKHx pacHGTax. KpoMe TOFO, paccMOTpennaa MexanHHecKaa MO^ejib y«:e ceftHac MOKCT 6biTb npHMeneHa ^,JIH anajinsa paspyinennfl B OTflejibHbix sjieMeHTax OTB6TCTBeHHbix CHCT6M, TaKHx KaK Tpy6bi 6ojibinoro flnaivreTpa H ccpepHHecKHe coPesyjibTaTbi nccjie^,OBaHHH BHe^penbi B yTie6Hbra npou,ecc H cocTaBji5iK)T coneKOTOpbix pas^ejiOB cneu,Kypca "YcTOH^HBOCTb ^ecpopMHpyeMbix HS pasynpoHHHiomHxcH MaTepHajiOBl!MarHCTepCKOH nporpaMMbi "MexaflecpopMHpyeMoro TBep,a,oro Tejia", nanpaBjienHe 010800 - - MexannKa H YpajibCKoro cpeflepajibnoro ynHBepCHTeTa HMenn nepsoro POCCHH B.H.Ejibij,HHa.

Hccjie^OBaHHH, npe^CTaBjiennbie B flHCcepTai^HOHHoii pa6oTe, jiHCb npn no/mep>KKe rpanTOB POOH (npoeKTbi 10 08-00135, 10 01 96018p_Ypaji_a, 13-08-00135) H MOJio^e^KHoro nayHHoro npoeKTa YpO PAH ^ H 1 HH 539.

Hcnojib3OBajiCH annapaT MaTeMaTHqecKOH TeopHH KaTacTpo(p, TeopHH oco6eHHOCT6H,n,H(pcpepeHHHpyeMbix OTo6pa>KeHHH, (pyHKH,HOHajibHoro anajiHsa H MexaHHKH flecpopMHpyeMoro TBep^oro Tejia. MeTo^ojiorH^ecKyio ocnoBy flnccepTauHOHHOH pa6oTbi cocTaBJi5HOT Tpyzi,bi Hayqnoro pyKOBO^HTejia, fl.cp.M.H., npocpeccopa B.B. CTpy>KanoBa.

IIojio:>KeHHa, BbiHocHMbie na 3am,HTy:

1. MeTO^HKa nocTpoeHHH e^HHoro BbinyKjio-BornyToro noTeHii,Hajia, OHHCbiBaK)uj,ero npn aKTHBHOM TpexocnoM pacTimenHH MaTepnajia ero CBOHCTBa KaK na CTa,n,HH ynpOHHeHHH, TaK H na CTa^HH pasynpoqneHHH.

2. MCTO^, onpe^ejieHHH qncjia peuienHH HejiKHennbix ypaBHeHHft paBHOBeCH5I rpaflHeHTHOH MexaHH^eCKOH CHCTCMbl.

3. HpHMeHeHHe M6TO^,a HbiOTona-KanTOpOBKHa ^JIM BbmncjieHHa nee/i,HHCTBenHbix penieHHH. npou,e/]ypa BbiGopa Heo6xoflHMoro nncjia naqajibHbix npnjiH>KeHHH fljia peajiH3au,HH HTeppau,HOHHOH cxeMbi HbioTona-KanTOpOEHqa.

4. HncjienHbiH MeTO^i, nocTpoennH cenapaTpncbi HOTeHH,HajibHOH 4)yHKu,HH MexannqecKOH CHCTeMbi H MeTO,fl,HKa onpe^ejienHH npe/i,ejibHbix snaqeHKH narpysoK.

H o6ocHOBaHHocTi> Hay^Hbix pesyjibTaxoB o6ecneCTporofi MaTCMaTHMecKOH nocTanoBKOH sa^aq;H, Hcnojib3yioui,eH MHqncjio flonynj,eHHH H KOppeKTHbiM npHMeneHHeM ^Jia ee peineHHH coBpeMeHHoro MaTCMaTHHecKoro annapaia H saKOHOB MexaHHKH ^,e(popMHpyeMoro TBepfloro Tejia, a TaKxe npOBeflenneM TecTOBbix pacqeTOB.

YcTaHOBJieno KaqecTBenHoe coBna^,eHHe pesyjibTaTOB, nojiyqennbix B pa6oTe, c pesyjibTaTaMH, npe^CTaBjiennbiMH B ny6jiHKau,HHx ^pyrnx JI6H HO paCTH>KeHHK) CTep^KHCBblX CHCT6M C paSynpOHHfllOIHHMHCJ H coBMecTHOMy pacTH»:eHHio c KpyqenneM KpyroBbix CTep^Kneft HS pasynpoqAnpo6aii;Ha pa6oTbi.

HOH pa6oTbi, o6cy5KflajiHCb H flOKjiaflbmajincb na cjie,n,yiOHj,Hx ceMHHapax H KOHpepeHUHflx: 16-an, 17-aa, 18-a5i, 19-aa BcepoccHHCKaa uiKOJia-KOHCpepeHi^HH MOJio/],bix ynenbix «MaT6MaTHHecKoe MOflejinpOBanne B ecTecTBennbix nayKax»

(r. ITepMb, 2006-2009); XV,XVI BcepoccHHCKaa SHMHHS uiKOjia no MexaHHKC cnjiouiHbix cpe,n, (r. IlepMb, 2009, 2011); III, IV, V PoccHHCKaa nayqnoTexHHqecKatf KOH(pepeHn,H5i «Pa3pymenHe, KOHTpojib H ^HarnocTHKa MaTepnaJIOB H KOHCTpyKii,HH» (r. EKaTepHH6ypr, 2007, 2009, 2011), mecTaa, ce^bMaa BcepoccnftCKaH KOH(pepeHHHH c «MaT6MaTHMecKoe MOflejinpOBanHe H KpaeBbie saflaqn» (r. CaMapa, 2007-2010), BcepoccHHCKaa Hayqno-TexHH^ecKaH KOH(pepeHij;HH «ripo6jieMbi 6esonacHOCTH KpHTHqHblX HH(ppaCTpyKTyp TeppHTOpHH H MyHHHHHajIbHblX o6pa3OBaHHH» (r.

EKaTepHn6ypr, 2007-2009); V BcepoccHHCKaa KOH(pepeHu,HH «MexaHHKa MHKMaTepnajiOB H paspymenHe* (r. EKaTepnnGypr, 2008); 37th International Summer School «Advanced Problems in Mechanics* (Russia, St.

Petersburg (Repino), 2011); Me^flynapoflnaa KoncpepennHfl no MaTeMaTHneCKOH TeopHH ynpaBjienna H MexanHKe (r. Cys^ajib, 2011, 2013); Me>K,n,yHaKOH(pepeHHH5i no,n,H(p(pepeHHHajibHbiM ypaBnennHM H flnnaMnqecKHM (r. Cys^ajib, 2012); BcepoccHHCKaa MOJio^ejKHaH KOHcpepeniniH "Conpo6jieMbi MexaHHKHl?(EKaTepHn6ypr, 2010-2012).

. Ilo TCMe ^HCcepTaiHiH ony6jiHKOBano 19 pa6oT (ne CHHTaa T63HCOB ^,OKJiaflOB), H3 HHX 5 CTaT6H B Be^yiUHX peH,eH3Hpy6MbIX HayHHblX H3flaHHflx, BXO/IHIHHX B nepe^enb HSflanHH, peKOMeHflOBannbix BAK.

O6i>eM H CTpyKTypa,a,JiccepTaii;piJi. ^HCcepTau,Ha COCTOHT HS BBe^eHHH, 4 rjias, saKjnoqeHHH, cnncKa jiHTepaTypbi, BKJiioqaiou^ero 151 HCTOHHHK, H npnjiojKeHHH. PaGoTa coflep>KHT 23 pncynKa H 4 TaGjiHHbi. O6m,Htt COCTaBJIJieT 115 CTpaHHII, ManiHHOHHCHOrO TCKCTa.

Bo BBe^eHHH o6ocHOBbiBaeTca aKTyajibnocTb TCMM HCCJieflOBaHHH H npno6tu,afl xapaKTepncTHKa pa6oTbi, npaKTHqecKaa H TeopeTHMecKaa snaJ)opMyjiHpyK)TC5i u,ejiH flHCcepTau,HH, onncbiBaiOTCH pesyjibTaTbi, BWHocuMbie na 3am,HTy.

B nepsoii rjiase npHBO^HTCH o6sop nayqHbix ny6jiHKau,Hft no nccjiefloBanHio TCJI, BbinojineHHbix HS MaTepnajiOB c BcJxpeKTOM pasynpo^HenHH. PacCMOTpenbi anajiHTHHecKHe H HHCjiennbie noflxo^bi K pemenHio KpaeBbix sa^a^ fl6(pOpMHpOBaHH5I TaKHX T6J1 Ha Ha^aiOIHeH B6TBH. BblHOJIHeH aHaJ!H3 OCHOBHblX nanpaBJienHH HCCJieflOBaHHH pa3ynpOHH5noLD,Hxc5i TCJI B cocTase ycToftqHBOH i. Ha ocHOBe anajiHsa npHBe^eHHbix CCWJIOK ccpopMyjinpOBanbi u,ejiH H flHCcepTan,HOHHoro nccjie^OBaHHH.

rjiasa nocBnu^ena onncaHHio O^HOH MaTeMaTHnecKOH MO^ejiH pa3ynpoqH5nom,erocfl Tejia, BSSTOH sa ocnoBy ^JIM HjijnocTpau,nn paspa6aTbiBaeMbix B xo^;e HCCJie^,OBaHHH MCTO^OB anajinsa nanpajKenHoro COCTOHHHH H MexannqecKaa CHCTCMa, B KOTOpoft Ky6nqeCKHH 3JieM6HT 4, BbinojiHeHHbiH H3 MaTepHajia, oGjia^aiomero 9cp(peKTOM ^,eopMau,HOHHoro pasynpoqHenHa, no^BepraeTCH TpexocnoMy pacTH^KenHK) B CHCTCMe, B KOTOpoft pacTHrHBaK)Uj,He ycHjiHa Ha Ky6 nepe^aiOTCH nocpe^CTBOM Tpex jiHHeftHO ynpyrHx CTep^Kneft c »cecTKOCT5iMH A i, A 2, A 3 cooTBeTCTBenno (pncynoK 1). B neflecpopMnpOBanHOM COCTO^HHH fljinna pe6ep Ky6a pasna e,n,HHHu,e (oTcqeTnaa KOHCJ^Hrypai^HH). Ppann Ky6a c O^HOH CTOponbi CKpenjienbi mapHHpaMH c a6cojnoTHO »cecTKHMH CTenxaMH, c flpyroft CTOponbi c ynpyrnMH CTep>KHHMH 1,2,3 TaKHM o6pasoM, HTo6bi B npou,ecce Tpexocnoro HH5i Ky6 Mor npHHHMaTb TOJibKO (popMy npHMoyrojibHoro CBo6o^Hbix KOHHOB CTep^Heii 1,2,3 sa^aiOTCH MOHOTOHHO BOSpacTaionepeMemeHHH u^u^u^ (>KecTKoe narpy^eHHe), jiH6o K HHM MOHOTOHHO BO3pacTaK)m,He ycHjiHfl P1? P2, -Ps (MHPKoe Harpy^KeHne). Harpy>KeHH6 CHCTCMbl HpOHCXOflHT HpH HOCTO5IHHOH TCMHepaType H CTOJlb MTO BO3MO2KHO HpeHe6p6Hb flHHaMHH6CKHMH 9(p(peKTaMH. B MO^GJIH BO3MO>KHbI H6CKOJIbKO BHflOB Harpy>K6HH5I CHCTGMbl, KOTOpblC MO^KHO pas^ejiHTb na Tpn THna: KHneMaTHnecKoe Harpyxenne nocpe^,CTBOM saflanHH Tpex nepeMeLQ,eHHH, cnjiOBoe narpy»:eHHe c noMOHj,bio npHJio^KennH Tpex CHJI, CMemaHHoe narpy^KeHHe nocpe^CTBOM o^Horo nepeMen^eHHH H flByx CHJI HJIH c noMOiHbio flByx nepeMeui,eHHH H oflHoft CHJIBI.

B xo^,e Harpy>KeHHa CHCTeMbi rpann Ky6a ne MCHHIOT opH6HTaii,Hio H B OTKOH(pHrypai;HH nojiynaiOT yfljiHHenHa i(i = 1, 2, 3). Ho rpanjiM Ky6a paBHOMepno pacnpe^ejieHHbie ycnjiHH HHTencHBHOCTbio qi. PaBnoSTHX ycHjinft paBHbi o~i = qiSia, r^e Sia - - njioma^H rpaneft B o~i MO>KHO paccMaTpHBaTb KaK HOMHHajibHbie HanpH^KCHHfl, paBHbie OTHOHI6HHIO B6JIHqHHbI CHJIbI, nepneHflHKyjIMpHOH K MaTepHajIbHOH HJIOm,a/],Ke, K ee nepBOHanajibHOH HJiom,aflH. BejiHHHHbi EI MO>KHO TpaKTOBaTb KaK ^,e(popMau,HH, onpe,n,eji5ieMbie OTHonienHeM yfljiHHeHHfl OTpesKa K ero nepBonaqajibHoii fljinne. CnHTaeM, qTO,zi,e(popMaiiHH Ky6a BOSpacTaiOT MOHOTOHHO, TO ecTb o6ecneqHBaeTCfl ero aKTHBHoe ^(popMnpOBanne.

Kbie CHCTeMbi HS Tpex Beui,ecTBeHHbix qnceji {EI, 2, 3} MO^KHO KaK 9JieMeHTbi TpexMepnoro eBKjiH^OBa npocTpaHCTBa fle(popMau,HH Rg, a ynopfl^OHeHHbie cncTCMbi Beui,ecTBeHHbix MHCCJI {cri, 0"2, cr^} — KaK BjieMCHTbi eBKJiH^OBa npocTpaHCTBa Hanpa^KeHHH ]R. Ilocpe^CTBOM neKOToporo TOlIKaM KpHBOH X (nyTb ^(pOpMHpOBaHHfl) CTaB5ITCa B COOTBeTCTBHG TOHKH B npocTpancTBe R^, oGpasyioinHe nyTb Harpy»ceHHa. ECJIH B HeKOTOpon TOHKe N(iN,2N,sN) ^e 3Ko6naH OTo6pa>KeHH5i (1) BbipoyKfl,en(detI = 0), TO corjiacHO TeopeMe o HCHBHOH cpyHKUHH peuieHne ypaBHennji x(e) = P B OKpecTHOCTH SN 3TOH TOHKH H6 HBJIHeTCH eflHHCTBeHHbIM flJIJI p 6 X($N), XOT5I OTO6pa>K6HH6 X OCTaeTCfl O,fl,HO3HaHHbIM. TaKH6 TOHKH HaSOBCM OCo6bIMH TOHKaMH aKTHBHbiM Harpy>K6HHeM y^eM noHHMaTb TaKOH npOD,ecc, npn KOTOPOM BJieMeHTapnaH pa6oTa 8 A nanpH^KenHH nojio»:HTejibHa, a HMenno, 6 A = p • 5e > 0. 3,a,ecb p — BCKTOP CHJIM B MOMGHT na^ajia ^orpyjKenHH, a florpy>K6HHe ocyHJ,ecTBJiaeTC5i nocpe^CTBOM sa^aHHH BeKTOpa ^e. Kor^a npn jiiooM aKTHBHOM /],orpy>KeHHH nojiHoe npHpanj,eHHe pa6oTbi AA BospacTaeT, TO conpOTHBjieHHe MaTepnajia yBejiHHHBaeTCH. Cjie^OBaTejibHO, MaTepnaji naxoflHTca B COCTOHHHH ynpoMHeHHH. IIojiHoe npHpau;eHHe A A BospacTaeT, Kor^a 6p-5e > 0. IIosTOMy ^annoe nepaBencTBO npe^CTaBJiaeT co6oii ycjiOBne o6m,ero ynpOHH6HH5I (ynpOHH6HH5I B HOIIOM). ECJIH HpH JIIoGoM aKTHBHOM Harpy}«eHHH BbinojiHfleTca HepaBGHCTBO ^p • Se < 0, TO HMeeT MGCTO o6nj,ee pasynpo^HeHne (pasynpoqHeHHe B HOJIOM). Kor^a flannoe nepaBencTBO cnpaBefljiHBo mnbKO neKOTOpbix nyTefi florpy>KeHH5i, TO 6yn,eM roBOpHTb, ^TO MaTepnaji naxoB COCTO5IHHH HaCTHHHOrO pa3ynpOHH6HHfl.

nojie p, o6pa3OBannoe OTo6pa>KeHHeM x; MO>KHO pasjio>KHTb na cyMMy BCKTOpOB p = c + r, r^,e BeKTOp c onpe^ejiaeT HOTeHu,HajibHoe BeKTOpnoe nojie (II — CKajiHpnbiH noTeni^Haji BeKTOpnoro HOJIH), a BGKTOP r 3a,n,aeT cojienoH^ajibHoe none. Bcrofly ^ajiee 6yn,eM nojiaraTb, ^TO ^ecpopMnpOBaHne HpOHCXOflHT B nOTeHHHajIbHOM HOJie C nOTeHD,HaJIOM n(i, 2, 3), TO eCTb OTO6pa»:eHHe x onpeflejiniOT (popMyjibi p = grad II.

, MTO B cjiyqae fle(J)OpMHpOBaHHfl: B noTenuHajibHOM nojie co CKanoTemjuajiOM pa6oTa Hanpa^KennH B flannoM cjiyqae onpe^ejiaeTca HOTeHH,HajibnoH {pyHKL^Hefi II. A ajieMeHTapHaa paGoTa Hanpa^KenHH pasna dA = deT -H(lT) -de, r^,e H(TV) ecTb Maxpni^a Fecce CMenianHbix nponsBo^Hbix noTeHD,Hajia II. OTKy^a cjie^,yeT, MTO ecjin B H3o6pa>KaiouieH TOHKC, HS KOTOpott npOHCxoflHT florpy^Kenne, reccnan (pyHKLi,HH II nojio>KHTejibHO onpe^ejien, TO cpyHKUHH II CTporo BbinyKJia BHHS, HTO OTBenaeT ycToft^HBOMy xapaKTepy npouecca flecpopMnpOBanHH (ynpOHHeHne). Kor^a npn florpyjKeHHH B jno6oM HanpaBJienHH reccnan OTpnuaTejibHO onpe^ejien, TO (pyHKH,HH II CTporo BbinyKjia BBepx, HTO OTsenaeT o6nj,eH HeycTOHHHBOCTH npon,ecca ^e(popMHpOBaHH5i (o6mee pasynpOHHeHne). ECJIH »ce reccnan SHaKOHeonpeflejien, TO (pyHKLi,HH II HG BbinyKjiaa H He BornyTaH, TO ecTb HMeeT MecTO ce^jiOBaa TOHKa. Ilpn ORHHX nyTHX ^,orpy>KeHH5i, HCXO^UIHX HS sa^annoH H3o6pa>Kaiouj;eH TO^KH, TeJIQ ynpo^HHeTCH, npn flpyrnx — pasynponniieTCH (qacTH^noe pasynpOHneHHe).

HTO COCTOHHH6 pa3ynpOHH6HHe 6CTb COCTOHHHe Co6cTB6HHOH H6yCTOHMaTepnajia ((pHsnqecKOH neycTOHqHBOCTH), npn KOTopofi nenpepbiBHOCTb npon,ecca flecpopMnpOBanna BO3MO>KHa TOJIBKO npn cnen,HajibHbix ycjioBH5IX narpy>KeHH5I, HO^aBJiaiOniHX flaHHyK) HeyCTOHMHBOCTb.

Hpn O,HHOOCHOM flecpopMHpOBaHHH nojiHafl ^HarpaMMa nojiyqacTCH fliKppepeHIIHpOBaHHeM O^HOMepHOH nOTeHLOiajIbHOH (pyHKHHH, HMeiOmeit o6jiaCTb BbinyKjiocTH BHHS, OTBeqaioiHyio ycTOHqHBOCTH MaTepnajia, Toqxy neperna, onpe^ejiHKtniyio norpannqnoe cocTOHHHe H oGjiacTb BbinyKjiocTH BBepx (HeycTOHqHBOCTb MaTepnajia). TaKHM o6pasoM, MOJKHO c flocTaToqHOH crreneHbio ToqnocTH cqHTaTb, qTO npn neo/],HoocHOM fle(popMHpOBaHHH c pasynpoqCBoiicTBa MaTepnajia ^OJI^KHW onncbmaTbCH MHoroMepnoH HOTeHii,HyHKu,HeH, HMeiouj,eH o6jiacTb BbinyKjiocTH BHHS H, B o6meM cjiyqae, ce/i,jiOBbie ToqKH (HJIH o6jiacTH BbinyKjiocTH BBepx). A Tax>Ke TOHKH, pas^ejifliouj,He COCTOHHHH ynpoqHeHHH H paaynpoqneHHH.

ECJIH cpe^a onncbiBaeTCH nenpepbiBnoflH4)4)epeHu,HpyeMOH(pyHKu,HeH II, TO B npocTpancTBe fle(|)OpMaH,HH npncyTCTByeT TOJIBKO O^HO cnjiOBoe nojie. B 3TOM cjiyqae Bee noBepxnocTH ypOBHH noT6Hu;Hajia II no^o6Hbi ynpyroro MaTepnajia H HMCIOT saMKHyTyio (popMy. Hepexofl B cocTOHHne njiacTHqHOCTH, a TaK^Ke flocTH5K6HHe npe^ejia npoqnocTH Tor^a, Kor^a yzjejibHaa noTeHH,HajibHaH sneprHH npHHHMaeT H6KOTopbie saflannbie snaqenna, ne 3aBHCHm,He OT BH,n,a nyTH flecpopMHpOBanna.

Ka>K/i,oMy TaKOMy COCTOHHHIO OTBeqaeT KOHKpeTHaa nosepxHOCTb ypOBH^, qTO comacyeTCH c snepreTHHecKOH Teopnefi npoqHOCTH BejibTpaMH-Xsnra. KpoMe Toro, cym,ecTByeT npeflejibnaa noBepxHOCTb ypoBHa, na KOTOpofi noTeHij,Haji HMeeT MaKCHMajibHoe SHaqenne, paBnoe paGoTe paspymeHHH, KOTOpaa npn O^HOOCHOM pacTH^KenHH pasna njiomaflH, orpannqeHHOH HOJIHOH ^narpaMMOH flecpopMHpOBanHH c naflaKtmefi RO uyjisi BeTBbio. Ha ocHOBannn STHX Hfleft B pa6oTe npHse^ena, ne npeTenflyji na o6uiHOCTb, o^na MeTOflHKa nocTpoenHH TaKHX noTeHHHajibHbix 4)yHKn,Hii, onncbiBaiom.Hx KaqecTBennbie xapaKTepncTHKH pa3ynpoqH5iiouj,eroc5i MaTepnajia.

CyTb ee saKJiioqaeTCH B cjieflyromeM: B ycjioBHHx Cannon nocTanoBKH sa^aqn MaTepnaji Ky6a aBjineTCH qacTHbiM cjiyqaeM HSOTPOHHOH oflHopOflHOH cpe^bi FenKH. KaK HSBCCTHO, npn HSOTepMHqecKOM TpexocnoM /i,e4)OpMHpOBaHHH 06pasua 3THX cpe^i, yflejibnaa HOTeHH,HajibHa5i aneprHH fle(popMaij,HH OTO>K^,eCTBJI5ieTC5I CO CBOGoflHOH 9HeprH6H Ha BC6X CTaflH^ MaTepnajia, BKJHonaH H aaKpnTHqecKyio era/niio (pasynpoqHeHne). Hs npeflnojio>KeHH5i BbiTeKaeT, qTO pa6oTa Hanpa>KeHHH ne saBHCHT OT Bn^a nyTH flecpopMHpOBaHHH, a onpe^ejiHCTca nojioxeHneM naqajibHOH H KoneqHOH ToqeK B npocTpancTBe fle(popMaH,HH.

PaccMOTpHM y^ejibnyio noTeHLi,HajibHyio Bneprnio ynpyrnx ^ecpopMauHH We Ky6a KaK KBa^paTHHHyio (popny. MaTpHii,y, OTBeHaK)m,yio STOH {popMe, MO>KHO paccMaTpHBaTb KaK MaTpHH,y CHMMeTpHqnoro npeo6pasoBaHH5i npocTpancTBa ^ecpopMaijHH, a snaHHT, ee MO>KHO npHBecTH K nyT6M opToroHajibHoro npeo6pasoBaHH5i 6asHca. B HOBOH cnereMe KoopTa MaTpnu,a onpe,n,eji5ieT noBepxHOCTb BTOporo CH ajrancoHflOM c H3B6CTHbiM cooTHOuieHHeM rjiaBHbix ocett. B CHjiy Hbix npe^,nojio>KeHHH noBepxHOCTH ypOBH5i e^HHoro noT6HLi,Hajia II B o6jiacTH neynpyrocTH TaK>Ke 6y^,yT sjiHncon^aMH c TGM xe OTHOineHHeM rjiaBHbix oceft.

OcTaeTCH TOJibKO naiiTH pacnpe^ejieHne SHaneHHH noTempiajia II no STHM noBepXHOCTHM ypOBHM. RjlK STOFO Ha O,fl,HOH H3 OC6H HOBOH CHCTCMbl KOOpflHHaT 6epeM npOHSBOjibnyio TOHKy H onpe^ejiaeM sjinncoH^,, HO KOTOPOM ona jie>KHT.

SaTGM, yHHTbiBaa, HTO nyTb, HanpaBJienHbift no STOH OCH, peajinsyeTca npn HHCTOM cflBHre, no nojiHon flnarpaMMe c^BHra Hafl^eM sneprHio fle(popMan,Hii.

H, naKoneu,, flejiaa oGpaTHyio saivteny KOOp^nnaT, nojiynaeM HCKOMbra cH sjieMGHTa 6biJi nojiynen cjie^yiomnH eflHHbiH noTeHii,Haji:

cooTBeTCTByion;aH ^aHHOH sanncn noBepxnocTb ecTb TpexnoBepxnocTb B neTbipexMepnoM npocTpancTBe. KanecTBeHnbiH BH/I; noCTpoenHoro noTenijHajia npn (pHKCHpOBanHofi oflnon nepeivieHHOH npnBeflen na pncyHKe 2. KpoMe TOFO, MaTpni],a TaHreHii,HajibHbix jKecTKOCTen, AaHHbiM HOTeHH.HajioM, HMeeT uiecTb He paBHbix Hyjiio KOMHOHCHT, HTO BaeT na npHo6peTeHHyio B xo^e neynpyroro MaTepnajia.

nocBam,ena npo6jieMe noncKa MCTOflOB, no3BOJi5nouj,Hx nanTH BCG BOSMO^KHbie pemeHHfl ypaBHGHHH paBHOB6CH5I. CHTyaH,HH, KOrfla COll CTOHHH6 HCKOTOpOH CHCT6MbI (rpaflH6HTHOH) OHpeflCJIHIOTCH (pyHKHHefi THna noTenHHajia V(Xi,yj), saBHCHmefi OT KOHeqnoro qncjia sa/jasaeMbix napaneTPOB ynpaBJIeHHH H napaMeTpOB COCTOHHHH, onpe^ejiniomHx nojio^KenHe CHCTCMbi npn sa^aHHbix ynpaBJiennnx, HBJIHIOTCH oqenb pacnpocTpaneHHbiMH. ECJIH, Toro, noTenimajibHaH (pyHKUHH HBJIHCTCH HeBbinyKjioft, TO ypaBHeHHH TaKHX CHCT6M MoryT HMeTb Hee^HHCTBCHHoe peuieHHe. CymecTByIOT TaKHe coBOKynnocTH napaMeTpOB ynpaBJIeHHH, KOTOPMM OTBeqaeT necKOJibKO paBHOBCCHblX COCTOHHHH CHCT6MbI. 3naHH6 XapaKT6pHCTHK BC6X HOJIO»CeHHH Heo6xo^HMO fljiH anajiHsa nanpn^KeHHoro COCTOHHHH HCCjie^yeMoft CHCTeMbi. KaK npaBHJio, sapanee nensBecTHO, cym,ecTByiOT JIH BOo6m,e pemeHHH, H ecjin cym,ecTByiOT, TO CKOJibKO.

CncTeMy ypaBHeHHH paBHOBecHH s^ecb H ^ajiee paccMaTpnBaeM KaK (pepeHti,HpyeMoe OTo6pa>KeHHe B o6m,eM cjiynae o6jiaflaiom,ee CBHsaHHbiMH c Bbipo>K^eHHeM MaTpHii,bi 5lKo6n. B flaHHoft rjiase na ocnoBe CBOHCTB TaKHX OTo6pa>KeHHH H3Jio>KeHa M6TO/],HKa onpe^ejieHHH HHCjia pemeHHH H HX BblHHCJieHHH C HOMOILI,bIO MHCJI6HHOH CX6MH HblOTOHa-KaHTOpOBHqa.

HSBCCTHO, HTO OCHOBHOH npo6jI6MOH HpHM6HeHHH MCTOfla HBJIH6TCH Bbl6op TaKHX naqajibHbix npn6jiH>KeHHH, HaqHHan c KOTopbix MGTOR CXO^HTCH, npnneM HHCJIO HanajibHbix npn6jiH>KeHHH ^OJI>KHO paBHHTbcn qncjiy pemeHHH. ^,JIH naxo^KfleHHH cooTBeTCTByioiHHx HanajibHbix npnGjiH^KeHKH HcnojibsyeTCH npejj,CTaBJienHe ypaBHeHHH paBHOBecHH KaK OTo6pa>KeHHH npocTpancTBa B npocTpancTBO ynpaBJIeHHH. BTO OflHoanaHHoe OTo6pa>KeHHe nosBOjineT B npocTpancTBe ynpaBJieHHH o6jiacTH, ^JIH ToqeK KOTopbix ypaBHeHHH paBHOBecHH HMCIOT oflHHaKOBoe qHCJio pemeHHH, H, KpOMe Toro, npoo6pasbi STHX o6jiacTeH B npocTpancTBe COCTOHHHH, B KOTopbix H cjie,n,yeT HCKaTb naqajibHbie npn6jiH>KeHHH fljin MeTO^a HbioTona-KaHTOpOEHqa. Paspa6aTbiBaeMbiH ajiroPHTM qncjieHHoro onpe^ejieHHH xapaKTepncTHK Bcex paBHOBecHbix COCTOHHHH (B TOM qncjie H HeycToftqHBbix) npOHJunocTpHpyeM na npnMepe napaMeTpOB paBHOBecHH CTep>KHeBOH MexanHnecKOH CHCTeMbi, m,eH TpexocHoe pacTH>K6HHe 3jieMeHTapnoro Ky6a HS HejiHHefiHoro pasynpoqHHiom,erocH MaTepnajia npn >KecTKOH CXGMC Harpy>K6HHH.

ITpn aKTHBHOM ^,e(popMHpOBaHHH HCCJie/i,yeMaH MexaHHHecKan CHCTeMa rpaH ee nose^eHHe onncbiBaeTCH noTeHii,HajibHOH nepBbie Tpn cjiaraeMbie — 3TO aneprHH fle(popMau,HH jinneHHO ynpyrnx CTep>KHeH 1,2,3, nocjie^nee cjiaraeMoe — BbinyKjio-BornyTbiH noTenn,Haji, onpeflejiHiom,HH CBOHCTBa Ky6a. IlapaMeTpbi ynpaBJIeHHH s^ecb HI -- nepeMernenHH CBo6oflHbix KOHII,OB ynpyrnx CTepxcHeH, napaMeTpbi COCTOHHHH E{ UHH Ky6a.

W,i = 0 onpe7j,ejMK)T OTo6pa>KeHHe / TpexMepnoro eBKJiHflOBa npocTpancTBa COCTOHHHH X B TpexMepnoe eBKJiHflOBO npocTpanCTBO ynpaBJIeHHH Y. TaK KaK ^aHHoe OTo6pa>KeHHe HMeeT oco6eHHOCTH, CBHsannbie c HeBbinyKjiocTbio noTeHunajia II, TO B neKOTOpbix TOHKax npocTpanCTBa COCTOHHHH MaTpHua HKo6n JiaHHoro OTo6pa>KeHH5i Bbipo^K^ena, npnqeM c TOHHOCTbio flo KOHCTaHTbi coBHa/i,aeT c reccnaHOM H(W} noTeHii;HyHKHHH CHCTCMbi. TOHKH, r^e MaTpnH,a 5lKo6n Bbipo^Kflena, o6pasyiOT B npocTpancTBe flecpopManHH MHorooGpaana KpnTH^ecKHx ToqeK OTo6pa>KeHH5i /, a HX o6pasbi B npocTpancTBe ynpaBjieHHii cocTaBjMiOT MHorooGpasne KpnTnqecKHx xHaneHKH OTo6pa>KeHH5i. B STOM cjiyqae OTo6pax:eHHe / OTHOCHTCH K KJiaccy GKjiaflbiBaioii],HxcH OToGpajKenHH. Ha pncyHKe 3 noKasana KapTHHa KPHTHHCCKHX SHaqennH CKjia^biBaromeroca OTo6pa>KeHHH /.

KPHTHHCCKHX TOHCK OTo6pa»:eHH5i / pasGnBaiOT npocTpanCTBO COCTOHHHH Ha HenepeceKaiomHecH OTKpbiTbie o6jiacTH, cocTaBjiHiomHe MHOJK6CTBO ^ = j^1,^2, W3}. OTo6pa5K6HH5I 3THX oGjiaCTeft B HpOCTpaHCTBO ynpaBJIeHHH oGpaayiOT MHO^KCCTBO $ = j^1, $2, $3}. TaK KaK OTo6pa>KeHHe / HMeeT oco6eHHocTH, TO o6jiacTH H3 $ nepeceKaiOTCH. Tor^a Ka>K,n,biH 3jieM6HT $s G $ ecTb oG'beflHHeHHe 4>s = M^|, r^,e &sk -- nepece^eHHH k 3JieM6HTOB HS MHo>KecTBa $. Tenepb, CCJIH TOHKa y € Y = M3 npHHa;j,jie>KHT HCKOTOPOH o6jiacTH $^, KOTOpaa, ecTecTBeHHO, HMeeT k npoo6pasoB B npocTpancTBe coCToaHHH, TO fljia nee paccMaTpHBaeMoe BeKTopnoe ypaBHenne HMCCT POBHO k peuieHHH.

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B pesyjibTaTe Bcex STHX onepaH,HH nojiynaeM npH6jin>KeHHbiH BH,H, cenapaTpncbi. HaKoneu;, onpefleji^H Ty nacTb cenapaTpncbi, npn nepeceHenHH KOTOpofi ocymecTBjiaeTca Bbixo^, HS o6jiacTH, orpaHH^eHHofl BeTB^MH cenapaTpncbi, onpeflejiHiOTCJi npe^ejibHbie narpysKH na CHCTeMy.

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2. HoKasaHO, qxo MexannqecKaa CHCTeMa c pasynpoqHflioiiniMCfl 9JieMeHTOM MOJK6T HMCTb HCCKOJIbKO HOJIOJKeHHH paBHOB6CHH npH SaflaHHOH BHeHJHeH narpysKe. Cos^ana M6TO/i,HKa, no3BOjiHK)Hj,aH onpe,n,ejiiiTb HHCJIO peuieHHti peUI6HHH ypaBHGHHH paBHOBeCHH rpaflH6HTHOH MexaHHqGCKOH CHCTGMbl.

3. PaspaGoTan MCTO^ Haxo>KfleHHH naqajibHbix npnGjiH^KeHHH, cooTBeTCTByIOLLI,HX qncjiy peuieHHft ypaBHeHHft paBHOBecna, ^JIH npHMeneHHH HTepainiOHHOH cxeMbi HbioTOHa-KaHTOpOBHqa, qxo HOSBOJIHJIO ncnojibsoBaTb MeTOfl HbioTona KaHTOpOBHqa ^,JIH Haxo^K^eHUJi Bcex peuieHHft.

4. npefljio>KeH HHCjieHHbift MeTOfl nocTpoeHHa cenapaTpucbi noTeHu,HajibHOH CpyHKUHH rpaflHeHTHOH ^HCKpeTHOH MexaHHqeCKOH CHCT6MbI.

5. FIpHBe^eHa MeTOflHKa HcnojibsoBanHH cenapaTpncbi fljia onpe^ejieHHa BejinqHH npe^ejibHbix narpysoK, npn,a,ocTH>KeHHH KOTopbix CHCTeMa paapyuiaTCH.

Pa3pa6oTaHHbie MeTO^HKH MoryT 6biTb nojiojKenbi B ocnoBy ajiropHTMOB fljia nporpaMMHbix KOMnjieKCOB, npe^HasnaneHnbix ^JIH pacqeTa necymefi cnoco6nocTH OT^ejibHbix ajieMeHTOB KOHCTpyKUHH, MaTepnaji KOTopbix o6jia^aeT PesyjibTaTbi, npe^CTaBjieHHbie B,n,HCcepTaifflH, MoryT cjiy>KHTb npn6jiH>KeHHeM HOBOFO no/pco^a K anajiHsy BonpocoB npOHHocTH pasjiHHHbix KOHCTpyKU,HH, KOTOpblH B CJI6flCTBHe yHCTa CTaflHH paSynpOHHeHHH HO3BOJIHT nojiHOCTbio Hcnojibsc-BaTb pecypc CiaTbH B >Kypnajiax, BxcyifliiiHX B CHHCOK BAK:

1. CTpyjKanoB, B.B. O6 O^HOM Mexo^e nocTpoemia e/jHHoro [TeKCTJ/B.B. CTpy>KaHOB, E.IO.IlpocBHpHKOB, H.B.BypMaineBa / Tejibnaa MexaHHKa cnjioniHbix cpe^,. — 2009. — T.2 JV5 2. — C. 96-107.

2. CTpy>KaHOB, B.B. YcTOHnHBOCTb ynpaBJieHHH rpaflneHTHbiMH [TeKCTJ/B.B. CTpy>KanoB, H.B.BypMameBa // BecTHHK TaM6oBCKoro CHTeTa. - -2011.- T.16 N°- 4.- C. 1183-1184.

3. CTpyjKanoB, B.B. BbiHHCjiHTejibHaa npou,e/i,ypa Haxo^KfleHHH Hbix napaMCTpOB Harpy>KeHH5i MexaHnnecKHX cucTeM [TeKCTJ/B.B.

HOB, H.B.BypMameBa // BbiHHCJiHTejibHaa MexaHHKa cnjiomnbix -2011.- T.4 JV* 4. C. 107-113.

4. CTpy^KanoB, B.B. MCTO^ HbioTona-KaHTOpOBMHa B sa^a^e 06 onpefleJI6HHH HeeflHHCTB6HHbIX peiljeHHH ypaBH6HHH paBHOBGCHH flHCKpeTHblX rpaflHMexaHH^ecKHx CHCTGM [TeKCTJ/B.B. CTpy>KaHOB, H.B.BypMaineBa // HncTHTyTa MaTeMaTHKH H MexannKH. — 2013.— T.19 JY^ 1.— C. 244-252.

5. CTpyjKanoB, B.B. MCTO^, HbiOTOHa-KaHTOpOBHMa B MaTeMaTHHecKOH MOTpexocHoro pacTH^KennH Ky6a us Maxepnajia c HeBbinyKjibiM noTeni^HajiOM [TeKCTJ/B.B. CTpy«;aHOB, H.B.BypMaineBa // BCCTHMK TaM6oBCKoro yHHBepT.18 N°- 5-2.- C. 2694-2695.

1. BypMauiesa, H.B. BncpypKai^HOHHbie MHO>KecTBa B sa^ane o TpexocnoM pacTH>KeHHH 3JieMeHTapnoro Ky6a [TeKCTJ/H.B. BypMainesa, B.B. CTpy>KaHOB // MaTeMaTH^iecKoe MOflejinpOBaHHe H KpaeBbie saflann: Tp. ^eTBepTofi Bcepocc. KOH(p. c Me>Kflynap. ynacTHeM. — CaMapa:CaMFTy. — 2007.— H.I. —c. 54CTpy»caHOB, B.B. O CBoiicTBax Ky6nHecKoro sjieMeiiTa npn ^KCCTKOM TpexocHOM ^.ecpopMnpoBaHHH [TeKCTJ/B.B. CTpy*aHOB, H.B. BypMameBa // c Me^AJHap. ynacTHCM. -- CaMapa:CaMFTy. — 2008.— H.I. —c. 301-308.

3. BypMauiesa, H.B. HTepan,HOHHbra npon,ecc pacneTa napaMeTpoB pasJK6CTKOM Harpy»ceHHH cHCTGMbi, peajiH3yioiii,eH TpexocHoe pacT5i>KeHHe Ky6a HS ynpyronjiacTHHecKoro pasynpOHHaiomeroca [TeKCTJ/H.B. BypMameBa, B.B. CTpy^KanoB// MaTeMaTHnecKoe HHe H KpaeBbie sa^anH: Tp. ce^bMoft Bcepocc. KOH(p. c Memflynap.

- CaMapa:CaMrTY. -2010.-H.1.-C.73-78.

4. Struzhanov, V.V. Newton-Kantorovich metho d of parametres' characterization of equilibrium of the system which realizes the triaxial strenght of the cube made from the nonlinear material with non-convex potencial [TeKCTJ/V.V. Struzhanov, N.V.Burmasheva// Proceedings of the XXXIX Summer School-Conference «Advanced Problems in Mechanics*.

St.Petersburg (Repino). -- St.Petersburg:IPME RAS. -2011.-p.468-476.

5. BypMauieBa, H.B. MCTO^, HbiOTOHa-KanTOpOBHqa pacqeTa Hee^HHCTBenHbix paBHOBecHH MexaHH^ecKOH CHCTeMbI, pacTflrHBaioiueH Ky6 HS pasyupoqMaTepnajia npn sa^anHH CHJI H nepeMeiu,euHH [TeKCTJ/H.B.

, B.B. CTpy»:aHOB// Tpyzjbi TOMCKOFO rocyflapCTBeuuoro SHKO-MaTeMaTHHecKan. II BcepoccHHCKan MOJio^e^Han uaynuafl KOHuocBHmeuuaji 50-jieTnio (pHSHKO-TexunqecKoro (paxyjibTeTa TOMCKOro rocy^apCTBCHHoro ynnBepCHTCTa «AKTyajibHbie npo6jieMbi coBpeMeuiiofi MCXaHHKH CUJIOIUHblX CpCfl H He6eCHOH MexaHHKH». TOMCK:H3,n,-BO ToMCKOrO yHHBepCHTeTa. —2012.—c. 76-80.

6. BypMauieBa, H.B. Hpe^ejibHbie SHaneHHa: uarpysoK npH CHJIOBOM H KHHeMaTHqecKOM Harpy^KennH CTep^HeBofi cucTCMbi, peajiHsyiomeii Tpexocuoe pacTH>KeHne Ky6a HS pasynpoquaiomerocH MaTepnajia [TeKCTJ/H.B. BypMauieBa, B.B. CTpyjKaHOB// MaTeMaTHHecKne MeTo^bi H MO^ejiH B TeopeTuqecKHx H npHKjia^,Hbix HCCJie^OBauHiix: c6. uayqu. Tp. [nop, uayqu. pe^,. P.A. TuMOcpeeBOH, fl-pa (pH3.-MaT. uayK H O.B. KyjiHKOBoft, Kaufl. ne/i,. uayKJ. -- EKaTepHHypr: HS^-BO Yp-ryHC. -2012.-BMn. 4(187)-c. 22-32.

7. CTpy>KaHOB, B.B. O6 O^HOH aa^ane ynpaBJieunH B MexannKe /i,e(popMHpOBauiia rpa^HeHTHbix CHCTCM [TeKCTJ/B.B. CTpy»:aHOB, H.B. BypMauieBa // «CoBpeMeuHbie npo6jieMbi MaT6MaTHKn». Teancbi 42oft BcepoccHHCKOH MOJIOemuoH uiKOJibi-KOHcpepeui^HH. — EKaTepHu6ypr:HMM YpO PAH. —2011.

c.14-16.

8. CTpyxauoB, B.B. MCTOA HbiOTOHa-KaHTOpOBHqa upu pacqeTe ycTOHqHBbix H HeycTOHHHBbix paBHOBecHH rpa^neuTHOH CHCTeMbI, ocymecTEjiaiomeH TpexocHoe pacTJDKeune sjieMeuTapuoro Ky6a [TeKCTJ/B.B. CTpy>KauoB, H.B.

BypMaiueBa // «MexaHHKa cujioiuubix cpe^ KaK ocnoBa coBpeMeuubix TCXHOjiorHfi» (XVII SHMHHH uiKOJia no MexauHKe cujioruubix cpe^,). Tesncbi flOKjia^OB BcepoccHHCKOH KouxpepeHUHH. HepMb: HMCC YpO PAH. —2011.—c.301.

9. BypMauieBa, H.B. BbiqncjieHHe napaMCTpOB uarpy>KeuH5i ^Ji5i ofluofi CTepjKueBOH rpaflneuTHOH MexaHHqecKOH cucTCMbi [TeKCTJ/H.B. BypMauieBa, B.B. CTpyxauoB// IV BcepoccHHCKaii uayHuo-TexHHHecKaH KOH(pepeuu,HH X IV uiKOJia MOJio/ibix yqeubix ceMHuap «Be3ouacuocTb KpnTHqecKHx uucppaCTpyKTyp H TeppHTopHH», MaTepuajibi KOucpepeHUHH H IUKOJIW. - - EKaTepuuypr: YpO PAH. -2011.—c. 104.

10. CTpy^KauoB, B.B. ^ecpopMnpOBanne HejinueHUbix rpa^neuTHbix CHCTCM:

, HeycTOHqHBOCTb, GncpypKauHH [TeKCTJ/B.B. CTpy^KauoB, H.B.

BypMameBa // TesncbiflOKjia^OBMe^KflynapoflnoH KoncpepenijHH no THqecKon TeopnH ynpasjienHH n MexanHKe. — M.:MHAH. —2011.—c.192-193.

11. BypMameBa, H.B. Hncjiennoe nocTpoenne cenapaTpncbi noTeHu,najibHOH (pyHKii,HH MexaHHqecKoft CHCTeMbi, ocymecTEJiHiOLneH Tpexocnoe pacTaKenne Ky6a, n onpe^ejienne npe^ejibHbix snaqennn napaneTpOB narpy>KeHHH [TeKCTJ/H.B. BypMameBa// Teancbi flOKJia^OB X BcepoccuficKoro c/besfla no (pynflaMenTajibHbiM npoGjieMaM TeopeTHqecKon n npHKJia^non MexannKn H BTOPOH BcepoccHHCKOft uiKOJibi MOJioflbix yqenbix-MexannKOB.

H.HoBropo^,:H3/i,-BO HH^Keropo^CKoro rocynHBepcHTeTa. 2011. —c. 23-24.

12. CTpy^anoB, B.B. Oco6enHOCTH OTo6pa>KeHHa CHCTeMbi ypaBnenHH c BbipojKflennoH MaTpnLi,eH 5lKo6n n npHMenenne MeTOfla HbiOTona-KaHTOpoBnqa onpe^ejiennH Bcexee peuiennn [TeKCTJ/B.B. CTpyxanoB, H.B. BypMameBa OKjia^OB Me^KflynapOflHOH Kon(pepeHn,HH no,n,H4)(pepeHn,HajibHbiM n ^HnaMHqecKHM cncTCMaM. — M.:MHAH. —2012.—c.162-163.