M E C H AN I K A TEOR ETYC Z N A I STOSOWAN A 4, 25, (1987)
PA3BHTHE HEJIHHEHHOft MEXAHHKH OEOJIO*IEK
B TPyflAX KA3AHCKOB IHKOJIBI
H . F . T E P E r yjiO B .:• .• :'• Ką sam, CCCP1. Ilpocfieccopy. X. M . M yurrapH ( 1900- 1981) H npotj)eccopy K. 3 .
( 1910 - 1986) Bbin ajia qecTB ocHOBani- iH Ka3aH CKoń uiKOJibi H ejiH ueH noft MexaHHKH 060-jionei- c, M iiorn e cJjyHflaMeHTajiŁHbie pe3yjiBTaTti KOTopoii B STOH o6jiacTH 3iiaH H H B 3OJIOTOH 4ioHfl; MHpoBOH H ayKH . I TepBon TaKoro pofla pa6oT oń ( 1938) SbiJia f(HCcepTar(HH X . M . M yirrcapH ,,H eKOTOpbie O6O6W,CHHSI xeopH H T O H K H X
c npHHOH<eHHHMH K 3a Aa ie ycToft^H BocTH yn p yr o r o paBH OBecH H " [ 1 , 2 ] , B K O T O P O H Bn ep Bwe B M H P OBOH H H TepaType 6&I JI H npoanaJiH 3H poBaH Bi cooTHOiueHHH TeopH H 060-jio^eK flJiH cjryiiaH n p o r n 6 o B , cpaBH H Mbix c TOJI I H H H OH oSon otiKH , H 6 Ł I J I H Bbin H caH bi KOppeKTHbie COOTHOIIieHHH 3TOH TeOpHH flJIH OpTOTpOIIHOrO H H 30TponH OrO JIHHeHHO yn p yr o r o Tejia. 3 T a n n o H ep c K an pa6oTa Bbi3Bajia IIOTOK HccJieHOBaiiHH B STOM H a n p a
B-. X B-. M B-. M yu iT apH B CBOH K ny6jiH Kai;H 5K [3 - 6] flaeT aH anH 3 n p e ^ e jio B n p n ~ npH6jiH>KeHHŁix TeopH H , ocH OBaH H bix Ha rH n oTe3ax KH pxrocJja- JlH Ba ( 1947) , BbiBOfl TO^H bix BBipa>KeHHit flJiH KOH e^H bix ffe^opMaqH H cpeflH H H oS noBepXHOCTH 060JI0t[KH B npOH3BOJlbHfcIX KOOpflHHaTaX ( 1948)
p y (1.1)
crpoH T ypaBHeHHH H ejiH H eił H oro K paeBoro atJx^eKTa. K i n c ji y (J)yHflaMeHTajibHbix H c c n e-cjieflyer c^irraTB ny6nHKaD(Hio X . M . M yn r r a p H ^Ka^tecTBeH H oe H CCJie^oH anpCCJie^oH CCJie^oH ceCCJie^oH CCJie^oH oro COCTOCCJie^oH CCJie^oH CCJie^oH CCJie^oH 3TipyroCCJie^oH OSOJI O^KCCJie^oH n p CCJie^oH M a n t ix pctyopniaujzCCJie^oHX. CCJie^oH n p o -H3BOJiBHbrx CM emeH H H x" [ 6] , B KOTopoii Bn epBbie 6bM a flana oSmienpH H H TaH T en ep b
TeopH H o6ojio^eK Ha cjia6biH , cpeAH H ił H CH U BH BIH H 3rH 6Bi H Ka-aH ajiH 3 K paeBoro stjupeicra.
B Tpyflax K . 3 . FanHMOBa B STH TOABI H H TC H C H BH O pa3BHBaiOTCH M C T O ^ M H ccicefloo6ojio^eK n p H KOHeqHLrx nepem em eH H H x H M an bix fle^opMaipKH X B qacTH n o c r -caMHX ypaBH eH H ft [7 - 9] H BapHartHOHHBie (J)opMyiiH poBKH 3ap,a.n TeopH H 060[10 H AP ]v 3 T H H M irorn e Ap yr n e pe3yjiBTaTbi X . M . M y m r a p n , K . 3 . F a jr a -MOBa H H X yqeHHKOB SBI JI H no«biTO>KeHBi B HX coBMecTH oił KH H re „ H ejiH H eiiH aH TeopH H yn p yr n x oSojio^ieK" [ 11] , B KOTopoii 6BI H o 6o 6m eH SoraTBin OIIBIT Ka3aHCKOH U I K O JI LI H ocHOBHbie pe3yjiBTaTbi, onySjiHKOBaHHŁie B M H P O BO H JiH TepaType K TOMy
EoJiBiuoe KamraecTBO H ay^H bix pe3yjibTaTOB B HejiHHeHHOH mexaH H Ke
542 M. T.
M . C , Ca^enKOBbiM A. B. H ApyrH MH , CCBIJIKH Ha KOTopbie MO>KHO B o 63o p a x [12 14]. 3 T H pe3yjib raTbi Sbijin noflbiTOHceHbi B KH H rax M . C . K o p
-[15], M . C . KopH H iiiH H a H <£>. C . H caH SaeBoft [ 16] , H . F . T eperyjiO Ba [ 17] , M . A . H jiBraMOBa [ 18] , M . A. H jibrawoBa, B . A. H BaH OBa, B. B. FyjiH H a [ 19] , H . B . C BH p cK o ro [ 20] .
H I H P O K O pe3yn bTaTbi yye H t ix Ka3aHCKOH uiKOJibi nejniHeHHOH MexaiiHKH o6ojioyei< B peryjiH pH o H 3flaBaeMbix „ H ccjieflOBarom x n o T eopn ii njiacrH H H OSOJIO-n p OSOJIO-n Ka3aH CKom rocyAapcrBeH H OM y OSOJIO-nH BepcH TeTe (Bbi OSOJIO-nycKH c I OSOJIO-n o XYU c 1962
no 1985 r.r.).
B KpaTKOM CO06meHHH HCf HMKEKOH B03JV10H<HOCTH OCBCTHTb BCe BOnpOCbl, KOTOpbie 6biJiH n o flBeprn yT bi H ccjienpBaH H io B Tpy/ jax Ka3aHCi<HX yyeH bix. E C JI H OTMCTHTB OCH OB-H bie OB-H anpaBJieOB-H OB-H OB-H OB-H ccneflOBaOB-H OB-H OB-H , TO K OB-HOB-HM a i e ^ y e r OTOB-HCCTOB-H cjieAyiomOB-H e.
O S Bo n p o c bi HenHHeHHoM Teopiffl yn pyrocTH n o nocTpoeH H io onpeflextH iom iix H o6ni;H e BapHau;HOHHbie npHHU,Hnbi 6bijiH H ccjieflOBaiibi B paSoTax K . 3 . [21 H flp.] H H . F . T eperyjiO Ba [22, 23 H p.p.]. B ny6jiH Kau(H n [24] ^aeTCH o 6o 6m eiiH e cjjopmyji ( 1.1) p,nx O6OJI OI
I K H i<ai< TpexM epH oro T en a.
B o n p o c b i nocTpoeH H H yTOiH eH H bix Teopnft miacTH H H o6ojiotiei< 6biHH n oflBepn iyT bi nccjieAOBaH H io B paSoTax X . M . M yixiTapn H H . P . T eperyjiO Ba [24, 2 5 ] , A. K . F a jiH H bin a, A. B . CayeiiKOBa c y*ieHHKaMH H p,p. B CTaTte [23] T eperyxrosbiM H . F . Bn e p -B t i e AaH a H enH H eiiH aH (J»opMyJiH
POBKa o Sm e r o BapHan,HOHHoro n pH H i^n n a, KOTOpbiii ceroflH H H a3biBaeTca npH H U iinoM Xy- BauiH ity.
M H orocjroH H bie H TpexcjioH H bie O6OJTC«KH 6bijiH o6i.eKTOM HccjieAOBaHHH X . M . M yiiiT ap si [ 2 6 ] , M . A . H n t raM O Ba, H . K, FajiH ivtoBa, A. F . TeperyjiOBa H ji,p.
^ n c jie iiH b ie MeTOflbi pemeH H H reoM eTpH ^ecKH H ejiH H eiiH brx 3ajxan pa3BHBajiHCB M . C . KopiiH iiiH H biM , <t>. C . H caH SaeBOH , M . C . FanH eBOH H
S a s a ^ a H 3rn 6a H ycroH iH BOCTH cpH3HHea<H
H 3 a ^ a u noji3yM:ecTH H 3yiiajiH ci. H . F . T eperyjioBbiM , M . C .
IIlHpOKHH K p yr 3aAa^I O COCpeflOTO^eHHblX BO3flei'lCTBHHX H KOHTaKTHOM B3aHMO-AencTBH H H ccjieflOBan B pa6oT ax K ) . I I . JKn rajiKo, H . F . T yptaH O Ba, K ) . I I , ApTio-XHHa.
A. B . Ca^enKOBBiM 6 WJ I pa3BH T yfla^H biii MeTOfl pem en H H 3aflaŁ
i CTaTHKH H flH H aM H K H o6ojioaM eK, noJiy^H BU iH ft Ha3BaHHe TeoperH KO 3KcnepH aMeH TajiLH oro aM ero ^a, yc n e u m o pa3BH TBiit H npH M eH eiiH biii K niH poKoiwy Kpyry n pH Kn a^H bix safla'q K ) . F . K o -H on jiesbiM -H flpyr-H M-H e r o y^ie-H-HKaM-H.
B MOHorpadpHH K . 3 . FanH MOBa [27] noABefleH H Tor e r o 6OJIBU IOH paooTbi n o n o c r p o eH H io HexiHHeHHOH MexaHHKH JiHHeHHO yn pyrH X oSon otteK c yqeTOM n o n e p e ^ H o r o C ABH -r a . B KH H -re [28] n o # H ayuH oń peAaK u n en K. 3 . FanHMOBa -rp yn n o ft aBxopoB H3Jio>KeHbi p ^ A pe3yjiBTaTOB n o TeopH H H npaKTH Ke pacqeTOB oSojioueK c y^ieTOM n o n ep e^ H bix C ABH T O B. B KH H re K . 3 . FanHMOBa n B. H . n a M y r a H H a [29] H3JiaraeTCH pa3BH Toe H M H H an paBJieH H e n o p a r a e i y oSoJio^eK CJIO>KHOH reoiweTpHH rpaH H i^bi odojio^KH . O 6 3THX H MHOrHX ApyrH X HCCJieAOBaHHHX M0>KH0 nOJiyiH Tb HH(J)OpMaUHK3 H3 o630pOB [ 1 2 , 14] , H yn oM H H yrbix Bbim e C 6O P H H K O B „H ccjieAOBaH H H n o TeopHH njiacTH H H 060Jio^iei< ". H axoflH TCJi B n e^aTH KH H ra K . 3 . F an H iwosa, B . H . riaSm yiiraH a, H . F . T e -p e r yji o sa ,,OcH OBaH i«c HeJiHHeiiHOH Teo-pH H o So jio qeK5
MEXAHHKA OBOJIOMEK B I<A3AHE 5 4 3
H 4>H3iraecKHe cooTHOiueHHH AiexaHHKH o6oJio^ieK [ 3 0 ] . H . C . F aH H eB n c c jieflyer n e jiH H efiH bie safla^H H 3r a 6a o 6o ^o ^eK c yuexoM nuacTH H ecKH x fletJiopMauH ftj a C . B . ^ e -peBam<H H c y^eHHKaMH H 3y^aeT n e c ym yio c n o c o Siio c T t OSOJIOH KH H 3 BOJIOKH H CTBIX KOMnO3H TOB.
2. O c o 6o cn eflyex OTMCTHTŁ oflHy M3 H an 6ojiee flpraix ny6jiHKaŁ(HH X . M . M yu iT ap H - ,Ka^ecTBeH H oe n ccjieflOBaH H e..." [ 6 ] , B KOTopoii # aH a KJiaccjKpHKau,H*i 3 a a a i TeopH H o6oJio»ieK. H e ocTaH aBJiH Bancb Ha Bcex fleraJinx STOH MaJioii n o oSteAiy, H O o^eH b c o gep>KaTejibHOH pa6oTBi, ocraHOBHMCH Ha oxfleJiLH tix so n p o c a x, n peAcraBJiH iom H X o c o -6biit H H Tepec. OrnpaBJiH H CB o x ypaBH eH H ił paBH OBecH H
qP = 0, (2.1)
= 0, (2.2)
H yCJIOBH H COBMeCTHOCTH fle<|)OpAiaU H H
^e^)}
= 0, (2.3)
— M
af,M
QJj- g.K = 0, (2.4)
H CBe^H Bee BejiimH H bi K 6e3pa3iwepH biM nyTeM yMHOHceHHH H a BeJiH ^H H bi Em
Ln
man — cooTBeTCTByioiuH e C Ten eim , E • — MOflym. K ) H r a , L — xapaKTepH bift n o n e -pa3M ep OSOJI O^K H H JI H n jiacxH H ti, X . M . M yu iT a p n n o ^ B e p r a e x ypaBH eH H H
TeopiiH yn p yr u x o So n o ^eK KaiecTseH H OM y aH anH 3y. B Ka^iecrBe ocH OBH oro n apaM eT pa, c KOTOP &IM cpaBH H BaroxcH n o p a ^ K H Bcex B C J I H ^ H H , B B O ^ H T C H n e K o xo p o e 3H a^eroie fle<J>opMai^H H yfljiHHeHHH ep, KOTopoe HMeex n o p a ^ O K ae( J) o
pHa n p e ^ e jie nponopu;H OH ajibH ocTH . BBefleH ne STOF O n apaM expa B K a ie c T Be 6a -30Boro onpaBAaH o xeM , *ITO B peajibH bix ycjioBHHX npuxoflnTCH H MCTŁ ASJI O c o6- beKTaMH, KOTopbie npejxH a3H aqeH bi P;JIH T o r o , ^ T O S W H ecxn HeKOTop&ie p e a n t H b i e H arpy3K H , y p o
-K o xo p t ix onpeflextnexcH flonycnaeMbiMH Be^HMHHaMH H anpjD -KeH H H H J I H ., nocjieflH H e H 3 K o xo p six HMeiox n o p n ^ o K sp. I I on biT K a on paB^aT b
T eo p m o oQojioieK npraiH TH eM npeAnojio>KeH H H o XOM , mo „SyA^M pacoviaTpH Baxb 6 e c -Majibie nepeiYiemeH H Ji", H e
Bbi^epwcH BaeT K P H T H K H , xaK KBK 3TO MO>Ker n p i i -K o^eH b HH3KOMy ypoBH io flonyci<aeMbix K paccMOTpeHHK) Ha ocH OBe TaKHX yp a
B-H arpy3OK B-H B-HanpB-H>KeB-HB-HO- fl;ec})opmB-HpoBaB-HB-Hoe COCTOB-H B-H B-H C, KOTopoe c BepHOCXbio omrcbiBaeTCH STH M H ypaBHeHHHMH, 6y^;eT oqeH b sajieKO ox peanBH O
B paSoxe paccMaTpH BaroTcsc ppz BH ^a H anpH H ceH H oro COCXOH H H H . I l e p B o e H3MeH5noinHMCH H fljiH H ero onepai^H H flH (j)(})epeH niH pOBaH jaH
H anpH >KeH H o- fle4)opM H poBaH H oe COCXOH H H C,
TO ecrb on epauH H flH c^dpepeH tiH poBaH H H H e M eH ser n opn flKa 43yHKi(HH. J ^ p yr o t i H anpH >KeH H oro COCTOH H H H TaKOB,
544 H . r .
rp;e X — noKa3aTejib H3M.eHHeM0CTH Hanpa>KeHHoro COCTOHHHJI. B npoijecce KaiecTBeH H oro aHauKraa cyH TaeTca, TTO
2Fh3
7 "*" = l E h A s ^ , M = ~ % V ,
H JIH n ocjie o6e3pa3MepHBaHHff
T " ~ hAafiey
e0Y, M* ~ A M ° * %r, ryje 2h — oxH eceH H aH K L Tojim n H a, T *13
H Mal>
— OTH eceH H tie K EL H EL2
ycvuw.fi
H MOMeHTbi, xap — yMHOHceHHEie n a L napajwexpbi H3MeireHHH K P H B H 3 I I , 3iiar< ~
yi<a-3b m a e x H a offHi- iaKOBOCTB n o pa^K O B conocraBJiH eM Bix BC JI H ^I KH . . I<jiaccH(|)HKaB[HH oC ojio^eK n o reoM eTproi n pefljioiKen a cjieflyioiqaH :
I< pH BH 3H bI,
p ]p KpBH 3H BI ,
K/ } ~ b ~ £ p'4
— n oJiorn e OSOHO^IKH,
ioc/J ~ b ~ 1 OSOJIOIIKH KOH e^IH Oit K p H BH 3H bI .
XapaKTep Ae4)opMHpOBaHiioro COCTOHHHH X. M . M yuiTapn KJiaccH({)nqHpyeT
KHIJHM o6pa30M (xxp ~ %, oia ~ co): •
« ~ 1 CHJIBHblft
H 3 r H 6 e H M eflJieH H O H 3MeH H K>meMCH H anpH >KeH H OM COCTOHHHH
, a ^ B
,n,jra oSojio^eK KOH CIH OH KP H BH 3H BI b ~ l(TH n la ) c norpeimiocTbio E p B c p a B H e H H H c eflH H H ijeft n pH xo flH M K o 6m e H 3Be c T H Ł iH ypaBH eH H H M JiH H eÓH OH O 6e3M 0M eH TH OM H anpK>KeH H OM COCTOHHHH
V
ar
a ?+ / = 0 , bipT ^ + g^ 0. (2.5)
,E[JIH n u a C T H H M ajIOń KpH BH 3H bI b <, Sp (THTI 1 B ) yCJIOBH H COBiWeCTHOCTH npH BOflH TCH
<**(*«?*?fietu = 0, c^V«^, , = 0, (2.6)
(2.7) Kai< ypaBH eH H H M ( 2. 5) n p n ^ = 0 MTOKHO C norpeuiH OCTBK) ep y^oB^eTBOpH TB, n o jia r a H
T«t) = cw- c^v^v, *. ( 2.8)
B m o r e ypasH eH H ii, peu iaro m ae s a ^ a ^ , npiiMyx
7 y_ v V 0 = O (2 91
MEXAHHKA OBOJIOMEK B KA3AHE 545 Afft %l - <T2 ) = aaap a"r + (1 - cr) cfa0 *, a — KoacJxJfflUHeHT I lya c c o H a .
Jinx o6oJioqi<H ManoH KpHBH3Hbi b ~ \ / sp ( r a n 1 c) ycJioBH H coBMecTH ocTH c r r o r p e m
-HOCTblO j/ fip H pH M yT BH fl
c^ V ( Va V^ Bw + bev Map) = 0 , {My *a? = 0 .
3flecb on aTb BO3MO>KHO npeflCTaBJieH ne ( 2. 7) , a n ep Bbie flBa ypaBH eH H H paBH OBecH H ( 2 . 5 i ) n pH gf = 0 yAOBJieTBopHiOTCH nocTaH OBKoń ( 2 . 8 ) . TaKH M o6pa3OM , B STOM cjiy-• qae umeeiw cHCTewy flByx H ejiH H eH H bix ypaBH eH H H BH ^a ( 2. 9) , ( 2. 10) .
H 3rn 6e H ~ | / ep H n p w MefljieHHom H3MeHeHHH H anpH >KeH H
oro COCTO-,a ~ *'> "A ź £p) A- JM O6OHOMKH KOHeiHOH KpHBH3HBI Ź> ~ 1 (THn
Tla) ypaBH eiiH H paBHOBecHH HMeiox B H S
V
ar
K" + ^ = 0, (ban + ^Tf + q = 0,
H AonycnaioT jiH H eapH 3ai^H io B nocjieflH eM ypaBH eH H H JiH iiib c n o rp eu iH o cT bio ]/ sp,
jiHHeapH3yiOTCH c norpeniH ocTBK) sp. p,o
r«e:
JI.JIH 06oJI0't
IKH M3JIOH KpHBH3HbI (Z) ;$ \ / Sp) (THII 2 B ) C nOrpeiUHOCTbK) Sp B CpaBHeHHH
c eflHHHi^eft ycJioBH H COBMCCTHOCTH Ko^am ^H ( 2 . 62) n n e p B t i e J\ B& ypaBH eH H H ( 2 . 5a)
no3BOjnHiOT BBecTH (pyHKUHio n p o r n S a >v H 4>yHKi?Hio ycHJiHH 0, Torfla I O K npHHHMaiOT BHfl
Af'^ V^ V V) + (b + )^ "^ VV0 + = 0,
= 0.
3 T H ypaBHeHHH H e yflaeTca jiH H eapH 30BaTb aa>Ke n p e H e 6 p e r a nOCOSCH H O H H TepecH bi BbiBOAbi n o Ka^ecTBeHHOMy aHajiH3y K paeBo ro I O K oi<a3ajiocb3 HejiHHeHHocTH B p a g e cjiy^aeB H rpaioT cymecTBeH H yio
I lycT b JI H H H H Xi = c o n st coBnaAaeT c JIH H H CM r p a H n n w cpeflH H H oft n o Bep xn o cT H H H e KacaeTCH acHMirroTH*iecKOH JI H H H H STOH n o sepxH o cT H H
$'> "Ó] m > 1, n ^ O, s > 1/ 4.
3T0M
546 H . T .
H 3 ypaBH eH H H H 3r a 6a ( 2.2) c u e ^ ye r , U TO
/? = m- n- 2s—t,
T o r ^ a i<aK H 3 ycnoBH H COBIACCTHOCTH RecJropMauHH ( 2.4) (ycJiOBHe F aycca) c jie a yc r ,
£ m m- n- 2s.
C onocTaBJiH H 3 T H flsa n oan eflH ue paseH CTBa, HMeeM
£ = m~ n- 2s, t = 0.
o6pa3OM , B 3OHe
4 M eM 6paH H sie .necjwpMauHH o ^H o ro n o p H / n o c H 3rn6H biMH . H 3 n e p B t i x flByx pasH OBecH H ( 2. 1) H ycjiOBH ń coBmecTHOCTH Koflai^qH ( 2. 3)
a ) j\ nsi UHJiHH.npHMecKHX H 6JI H 3K H X K H H M o6ojioqeK
Tn ~ r
1 34 ~ T
22s
2 p s,
6) AJ1H o6pa30M , n p n s ^ 1/ 4KOHCTHOH KpHBH3HbI b ~ 1 (/? = 0) n p H fia/ J ~ ep (fft = 1) H h ~ ep ( T O H I O H o6ojioyi< a) HMeeM
/5 = m—n—2s = 0, m—n—t = 1.
OTKy^a B CHJiy ?M = 1, f = 0 c^e,n;yeT, ^ T O n — 0,s = 1/ 2. TaKHM o6pa3OM,
B O6OH X c jiyq a n x c norpeniH ocn> K> H e 6ojiee yep B ypaBH eH H ax MO>KHO H OJIOJKH TB ^22 + ^22 — ^22 H T e
M CaMfclM. JIHHeapH3OBaTb K ypaBHeHHH B yCHJIHHX H 3e(J)0pMat(H SX. O T M C T H M , ^ T O B OSOH X c jiy^ a a x
2
H BBipa>KeHHe fl;e4>opManH H q e p e 3 nepeM emeH H H ^OJI>KH O 6BI TB coxpaH eH O B H OM
B cjiyt iae n o ji o r o ń oGojionun b ~ e *'4
n p n h ~ ep (TOHKaa o6ojioiiKa) nony^H M n = 0 ,
5 = 3 / 8 :
a
) ^22 ~ £ p '8
flJiH H euH jiH H flpiraecKH x o6ojioyeK H K22jb22 ~ ep18
, H 3r n 6a o ie ^ ye T n n c a T t B HejiHHeftHOM BH ^e ( 2. 2) , TaK Kan
nopH flKa sp la
B cpaBHeHHH c eflMKHqeft BHeceT So jit u iyio norpeuiH OCTBj 6) X2i ~ £ p/ 4
PRO. I^HJlHHflpH^eCKHX 060JI0^eK H X22/ b22 ~ |/ fip TaKHM 06pa30M , B 3TOM c n yya e JiH H eapH 3aqH H ypaBH em iH BO3MO5KHa c n o r p eu iH o c T t io | / ep. O T M C T H M ,
q xo JiHHeapH3aE(HH BŁ ipan<eHHH el t M epe3 nepeM emeH H H B03M0H<Ha JIH U IB C n o r p e u i -HOCTBK) fii'4
MEXAHHKA OEOJIOIEK B KA3AHE 547
o6pa3OM, B o6meM cntytiae sa ga n a o KpaeBOM 3cj)(|>eKTe 0Ka3biBaeTca H eH H OH .
B 1965 rofly B craTbe [31] Swjia flaH a KJiaccHcfiHKai^HH 3aAa*i xeo p n H B KaqecTBe Maxroro n apaiweipa H cnoJib3OBanacb
& = max
B 1972 r . B paSo're [32] 3xa i<JiaccH<f)HKauHH Swjia pa3BH Ta BBefleHHeiH e t q e OAH oro n a -paM eTpa H
' h
I — fxflvaisi BOJiHbi H3neHeHHH H anpH H ceH H oro COCTOH H H H . O^eBH flH O, ^ T O STO B03Mo>KH0CTt B^TOKH TB B npeAJio>KeHHyK) KJiaccH(J)HKaqHK) H HBJieHHH T u n a K p a eBo r o 3(|)4)eKTa. O npeH M ym ecTBax n epefl 3TH M H KJiaccH(|)HKai|HHMH BBC AC H H H KaK ocH OBH oro
Jioro rrapaM eTpa BejiH<- iHHŁ>i eap ~ sp ci<a3aHO Bbiiu e.
B 1980 r o s y B craTbe [33] B . IleTpaiiiKeBH MeM npefljio>KeH o p a c m n p e i ł n e KJiaccH (|)H xapaKTepa H 3rH 6aiiH H , paccM cn peH H oro X . M . M y m i a p n , H npeflJiOM ceno C M H -TaTB co ^ O ( 02 ) — M anoe to = 0 ( 0 ) — cpefln ee to — 0 (\ / &) — So Jt t u io e to = 0( 1) — K on e^H oe Jm<aił H H n p « &2
~ ep npeflCTaBJiH ercH KoppeKTH oń H BBefleH ae 6o jn .iiiero
ycjioBneM w = 0{}/ W ) — 0( e */ 4
) Bbi^ejiH eT B OCOSBIH KJiacc n o n o r n e 060-JIO^KH n p H HX 60JIbIU0M H 3rH 6e.
3 . B nocjieflH H e rojx,H B Ka3aH H pa3BH BaercH TeopH H n o c T p o e m o i
cooTH omeH H H flJiH aH H 3OTponH bix oSoJio^eK, B qacTHOCTH, fljiH KOMno3HTHŁix, o 6p a -30BaH H bix H3 MOH ocnoeB oflHOHanpaBJieHHO apM H poBaH H oii crp yK T yp bi [ 22] . 3 T H M H HCCjieflOBaHHHMH BOcnojiHHeTCH n p o 6eit B HejiHHeHHoft iwexaHHKe TOH K H X oCojio^ieK B nacTH n ocrpoeH H H o6m eH TeopH H A- ns o n p e ^ e n H i o m n x (4)H 3H iecKH x) C OOTH OH IC H H H , T o r^ a KaK B o n p o c t i TeopH H reoM eTpH qecKoft HejiHHeiiHOCTH H a cero ^H H ^ocTaToiH O rnySoKO pa3pa6oTaH bi. B ^ a c r n pa3BHTHH HccjieAOBaHHH H ejiH H eH H bix 3afl;a^, cBH 3aH -H t ix c KO-He-H-HbiM-H Bpauie-H -H -H M-H , cneflyeT OTMeT-HTb ycn ex-H IIojiBCKO-H uiKOJibi M exa-H exa-H K exa-H o6ojioqeK BO rjiaBe c n po(J). B . n e r p a u iK e B exa-H ie M .
a. H a n oBepxH ocrH npH BefleH H a So B He# ecpopMHpoBaHHoivi COCTOH H H H BBefteM C H C -Tejwy jiarpaH H <eBbix Koop^H H ax ^ "( a = 1, 2) c MerpEreecKHM TeH 3opoM a^ = (pa' Pp) H KOOpflHHaTHblMH BeKTOpaMH pa — dp/ dx*. H a 3KBHflHCTaHTHŁIX n o HOpMaJIBHOił K So
Z = X3
nOBepXHOCTHX Sz C OpTOM HOpM3JlH m KOOpflHHaTBI X°* HMOOT MeX-TeH 3op gap = ( ra • rp), r = p+zm, H KOopAHHaTHbie [BeKTopbr ra = 8r/ dxa
I I p H 3TOM g33 = 1, ga3 = 0. B fleCJJOpMH pOBaH H OM COCTOHHHH Ha S$ HMeeM a*p =
= (P *"P *). pS = de*/ dx
a, p* — p+u, a Ha S* HMeeM
Sh = W ' rf), r«* m 8r*/ 8x\ r* =
548 H . T .
XapaKTepH CTH Ka flecIpopMaijH H F p n ir a
8
U- (g*k- glk)/ 2
OTH eceH a K H a^aJitH OM y COCTOSJHHIO H o6pa3yeT TeH 3op
T e H 3 o p HanpHH<eHHii K O H I H Z1
HMeeT cocraBJiH iomH e ai fc
B ^e^opM npoBaH H OM 6a
a TeH 3op yaio BH L ix H anpH H teH H H T uuvieer B He/ i;ecpopMHpoBaHHOM 6a3H ce r< cocraBJiH -tik
H T = tlk
rtrk n p H BBinojiHeHHH ycjioBH ft / 'fc
= crt k
/ c*, » rfle g^;i—iyioTH OcTb B 3e4)opMH poBaH H OM cocTOHHHH. B cTaTŁe [22] n oKa3aH o3 • mT
O B cjiy^ae rtoji-H o r o pacceO B cjiy^ae rtoji-H O B cjiy^ae rtoji-H n a: B Bnp,e Tenira p a S o r b i BO B cjiy^ae rtoji-HyTpeO B cjiy^ae rtoji-HO B cjiy^ae rtoji-HO B cjiy^ae rtoji-HX O B cjiy^ae rtoji-HanpO B cjiy^ae rtoji-H>KeO B cjiy^ae rtoji-HO B cjiy^ae rtoji-HO B cjiy^ae rtoji-H O B cjiy^ae rtoji-Ha npO B cjiy^ae rtoji-H pameO B cjiy^ae rtoji-H O B cjiy^ae rtoji-H O B cjiy^ae rtoji-H x RC-n ox RC-n 3yqecT H de\ f? H x RC-njiacTH ^ecKH x fle(j)opMar(H H de\ l)
HMeeT wecTO 0606-4>opMyji
delk
5eik — deik— de\ ]? — ds^, eik—noxtH we flec^opMai^H H
, F-detk — n p n p am eH H H yn p yr t i x .gecpopM ainiH . 3flecB 3HaK <5(...) yKa3biBaeT Ha T O ,
n pH paiu eH H e cooTBeTCTByromeH BejiH ^H H bl H e e c n . ITOJIHBIH fl;n4)4)epeH i;H aJi, a n ojiH oe n pu pam eH Bt e cooTBercTByioineH Ben H iH H bi 3aBHCHT OT nyTH , n o KoiopoM y npoTeKaeT n p o u e c c .
( 3 . 1) , o C o m aio iu an cpopM yuy F pH H a H a cn yqafl H a n n q a H H e yn p yr a x n p n -fiopMainiH H cJieayeT H3 T o r o , ^ T O AJI H njioTHOCTH cBoSoflH oft 3H eprH H F a cjiy^iae itajimma. TemiOBbix n p o n e c c o B HMeeM
QtdF = a
ikde
tk- Q*sdT - jJJ P
Jd
Xj,
-
:• • • •
• Td - - v dt+YM -
( 1 2 >• • !'. . t • • • ;.. • . • c i : o n i .. • • , : • • [ q ^ y i r i i . . . • . - ; , .. !• :.
s — nnoTHOCTB 3H Tpon raij T — aScomoTH aji TeM nepaTypa, r—H H TeH CH BH ocn> . T e n n a , q = g^ r f — BeKTop noTOKa T eim a, PJ
— cocraBJiH ioiH H e o6o6iqeH -H BI X C-H JIJ %j — oSoSrą e-H -H we nepeM etne-H -H fl, -H a KOTopux C-H KBI P' npo-H3BOfl-HT pa6oT y, pacceH BaeAiyro B BH ^e xe n n a . B HccneflOBaHHH [34] n o K asan o , MTO H e MeHee 9 0 % M exa-H iraecKOexa-H paG oTŁi n a njiacTexa-H ^ecKexa-H x «ec|)opManexa-H exa-H x nepexoflexa-H T B TenjioByra
T a K KaK MexanH 3M fle<J)OpMaiiH H n p H rroji3ytiecTH ofltiaKos c MexaHH3MOM K o r o fletJiopMH poBaH H H , TO H n p H nojrey^iecTH cjie^yeT o>KH,n;aTŁ n e p e xo ^ a paSoTbi
HanpH>KeHHH H a necJiopmaiCHHx noji3yiiecTH B T erm o. TaKHM o6pa3OM , M exaH H qecKaa: a n e p r n H aik
de\ ^ H SH ep r n n aik
de$ n epexo^H T B T en iro. T o r fla corjiacH O ( 3.2)
M E X A H H K A OBOJICfflBK B I<A3AHE 549
rpp npHHHTO
H 3 cK33anHoro cjie^yeT (1).
B. OyHKUHJi F 3aBHCHT OT HHBapHaHTOB fle^opMH poBaH H oro COCTOHHHH, TeAinepa-Typw T H , mo>i<eT 6BITŁJ OT Apyrux napaMCTpoB npoijecca. B Ka^ecTBe
3e4)opMHpoBaHHoro COCTOHHHH Bbi6epeiw BenmiHHbi [22]
I2 = eape^>
h
= el
/l(f)ga/? = 0, ya = 2ea3. KacaTenBHan K iSz njiocKOcrt ecTB nJiocnocTB
CBOHCTB. B 3TOM cjiyqae flJiH tm coraacH o (3.1) imeeM 3F Qt* =
(3- 3)
(3.4)
H B ocax 3«ecb ta = ta3, Ax — Q3Fj8It. Rsin opTOTponHoro
OpTOTpOnHH A12 = 0.
Jinn TOHKHX oSojioieK t33
<4,tll
,t22
H , cne^oBaTejiBHO, 8Flde33
^ 0. TaKHM 06-pa3OM, F OT e33 He 3aBHCHT. r t o n a r a n nonepeMHyio cABHroByio
(y
a~ 0)
B OCHX OpTOTpOIIH H flJIH JIOKaJIbH blX KOOpAH H aT C XapaKTepH C TH KaM H glx — g22 = U
gl2 = 0 HMeeiw An = - A22 = 1, A12 = 0
Ix**2elx, I3 = 2e22, I2 = 1e\2.
IlpHMeM, IITO CHcreMa HanpH>KeHHii / "^ paBHOBecHan, npH pameiiH H deal!
BO3MO>KH bie.
B CHuy Toro, I T O deaft — fle(J)opiviaqH H yn pyrn e npH gdt
al1
— Aa^oydee
• dtafdetlfi> 0 .
- ex, e22 = e2, eiz = e3. M
a-(J)opMy. KO3(J)4)Hu;HeHTOB Alk n o po K ^aer noitojKHTejibHO onpe^eJieHHyio \ \ Alk\ \ Ex — ycJiOBHbie E2, 3
.
.550 H . T.
E
ltE
2, E
2~ E
2, E
3E
2, E
3- V
2M aTpH i;ti H ^ H flaer p,ikEt = JikiEk\ i,k = 1 , 2 , 3 (H e cyjwMHpo-a yc a o BH e nojioH ttrrejibH oft onpeoeneH H OCTH KBaApaTH ^H on (|)opM bi £ dtk det dek
ik
— K ycJioBHHM E, > 0 , ftikfiti < 1; / , k 1, 2 , 3 (H e cyiwMHpoBaTt) H a n p H M ep , H3 yc n o -B H H / tiia/ iai < 1, fJ,12£i = ("21^2 c n c a yeT ,
noJiymH M, fl12 < ł ?
n o3o6H Bie o u e m a i fljia Bcex qjieHOB MaTpmi;Bi 3an n c aT b c jie^ yio m ee cooTH OuieH H e
1,
r\ , rf,
x2 r]2 o6pa3OM , ecriH B, = QdF/ deit T Oż '
8B
2n p n ?; ->• 0 HMeeM, ^ T O cymecrBeH H BiM 6yfler JIH IIIB OAH H
BB
1lde
l~E
1.
x - > 0 ^aeT cym ecrBeH H Bie
(3.5)
( 3. 5) cooTBercTByer H H TH H OH MOflejiH o^H OH anpaBJieH H o apM H poBaH H oro CJIOH . ii ( 3. 6) cooTBeTCTByex opToroHajiBHO apMHpoBaHHOMy CJIOIO C H yjieBoń cflBHTOBoń cxeKJionjiacTKKOB oAH OH anpaBjieH H oro apMHpoBaHHH B H anpaBJieH H E O C H Qxi.; Et a 4600 M i l a , ^2 ~ 18000 M i l a , G12 <X 4500 M i l a H , cneflOBaTenbH O,
r]2
~ 1/ 3, x2
~ 1/ 3, 7] ~ x.
X O T H 1/ 3 H H e M ano B cpaBH eH H H c e^H H H ueiij oflHaKO B c iu iy acH M irroTH ^ecKoro xofla n o Majiojwy n a p a M e r p y npH BefleH H Bie orł eHKH KOppeKTHBi.
JXns rj ~ x H3 ( 3) cue^yeT j MTO dB^jde^ B acHMirroTirqecKOM pa3JioH<eHHH H aerrcH c H Jien oB, co^ep>Kait(H X rj B H y^eBoft cren eH H , dBi/ de2 — c qjieH
n ep BBie c r en eH H rj, 8Bxj8ez — c qjieH OB, coAep>Kaiu(HX KBaflpaTBi rj.
MEXAHHKA OEOJIOIEK B KA3AHE 5 5 1
3;ą ecB q>2 B acHMm- oTHqecKOM pa3JioH<eHHH n o rj na^H H aeTCH c yjien o B, coflep> Kam n x B H yjieBoii cren eH H : k=0
B
2= W i O O + ^ Va fc i. e
2) + rf^
3{e
x,e
2,e
3,r])
e
i,e
2., e
3,rj)
B acHMiiTOTiraecKOM pa3Jio>KeHHH= y)
3k=
^
4 k= 0 npn k > 1. H 3
c y q e r o M T o r o , ^ T O n p w ei2 flecJ>opMaitH H e n , e22 H e At o r yr Bbi3BaT b H a n p H ł K e H H H ti 2,
0*22 = - #21 ( e n . ^22)^11 + - ^22(^11,
E C J I H B acH MirroTirqecKH X pa3JioH<eHHHx jjjin JSX H B2 orpaHHqHTBCH , TO on peflejiH iom H e co o T H o m em ra flOJi>KH bi 6BI TB npeflCTaBJieH bi B
e^ii = S11(en)en+£l2(e11)e22,
Qt22 = JB2i ( e i i ) e
B npocreH iueivi c jiy^ ae, K o r ^ a B pa3Jio>KeHHH AJI H BX coxpaH H ercH O,II;HH qjieH , TO ec T t - Si = V O ^ O J onpeAexiH iomH e cooTHOineHHH npH BOAH Tca K
coBnaflaroT c COOTH OU ICH H H MH , npe^jiroKeH H tiM H B paSoTe [ 3 5 ] . c . Oim uieM n p o i t e ^ yp y 3KcnepH MeH TajiBH oro o n p e^ e^ eH H H cjyyHKą im At(Ij) orpaH iweH H H X ya ~ 0 , de\ p = 0 , deffi = 0 , T . e. B c jiyq a e M ajibix n o n e p e ^ H b i x
r o B. Jlflu ycH jim ł , OTHeceHHbix K H aqanbH OMy HeflecpopMHpoBaHHOMy COCTOH H H IO H B Hefle<J>opMHpoBaHHOH
f f J
- h - h - h h, T
12= J
et
12dz, (3.7)
hrp,c 2h — ToranH H a o6ojioqKH . ITycTB i^H JiH H flpH iecKaa o So jio ^ n a n3roTOBJieH a Koń OAHOHanpasjieHHO apMHpoBaHHOił jieHToft n p n y r a a x HaiwoTKH <pk— ± <P*
r
f le
9 * — yr jibi, KOTOpbie o6pa3yioT H anpaB^eH H H apM H poaaH H H c o6pa3yK> m eii
552 H . T. TEPETYJIOB
KOJiH ^ecTBe cjioeB B O5OKX H anpaBJieH H H X. r i p n OTOM flJis cjiy^iaH e12 = 0 H3
(3.3)
If = 2(eilcos1
<p + e22s'mz <p), I3 = 2(ell 21? = (en~e22)2 sm2 2(p, A\x = ~ A\2 = c o s 2 ^ , A\2 - si T ai< i<aK I j OT HOMepa CJIOH H e aaBHCOT, TO coraacH O ( 3. 4) , ( 3.7)
Ttl = 2h[A,_ • ( l + c o s2 < p ) + ^3 • ( 1- co s295) ]+ 2hA2- (en- e22)siii 2
2f, T22 =2h[A1(l- cos2<p)+A3(l+cos2<py]- 2hA2(e11- e2Z)sin
2
2<p. ®® Jlna on pe«eJieiiH H A2(l2) OTCIOAS n p n cp = ?r/ 4 nojiy^H M (J)opMyjiy
4hA2(e22~e11) = TZ2- TXX ( 3.9) B n p eAejia x TO^H OC TH pacraioTpeH H oro paH ee acH M rrroTimecKoro rrpH 6jin>KeH H a A2
3aBHCHT JIHIHB OT I2 > flJIH KOTOpOrO n p H fp = jr/ 4 HMeeM 2/2 = (e11- e22)
2
. I I p H T1X—T22 > 0 H 3 ( 3.9) AJI H A2 nojiy^H M
A
2(I
2) = ^ i ^ Ł
.
4/ JJ/ 2/2
B npe,n;ejrax TOH »ce acHMirroTHiiecKOH T O I H O C T H AX = A^Ix,!^, A3 = Az{Ix,I3). on pe^eJieH U H STH X ^ymax,ii& H3 ( 3.8) n p a cp ^ njA H H 3BCCTH OH ć pyKKaym A2(Iz)
= Tncos 2 (p—T22sm 2 (p—2hA2(eXi. — e 2 2 ) q , 4hA3cos 2 <p = - T11sin i <p+T22cos 2 <p+2hA2(.en- e22)sm 2 2<p. ' 3 fl e c t Bee BxoAH mH e H BH O en, e22, h cJieflyeT Btipa3H TB Tiepe3 It H 73 coraacH O ( 3. 3) ,
TO e c r b 26^00521)? = Iy cos,2 q — 73 sin2 < p, 2e2 2c o s2c > = 73c o s 2 c > —Xt sin2 ^ , ^cp At, A2, CBH3B MOKfly 3aMepHeMBIMH B 3KCHepHMeHTe A tk i- ds,. , ds~ ds2
H BeJIHUHHaMH eu , ^22 , KOTOpaH flaeTCH B
C flpyroft CTopoH bi, ecjiH K i;njiH H flpy npHJio>KeHa o c eBa n pacTHrHBaK>maH cn jia P H BH y-TpeH H ee flaBjieH H e q, TO n o n H a n o ceBaa cH Jia n p H 3aKpBiTfcix flH mn;ax Q = PA- nR^q CO3flaeT B I^HJIHHflpe HCTHHHBie HanpH>KeHHH
^ £,a Q S tu Q & * • " • ' ' " I '1
MEXAH H KA OBOJIOIBK B KA3AH E 553
3flect i?* = R(l+d
2), K = h(l+A
3). C yqeTOM nepexo^a
H3 BapH aijuoH H oro ypaBHeHHH n p H im n n a B03M0>KHbix nepeM emeH H ii n p n ypaBHeHHH
3flecB P J — BeKTop Hanpn>KeHHH Ha cpe3e c rpaH H ueft 5 *. O rcio;ja c n e ^ ye r , MTO B cjry-yae «j = 1 ,n2 = 0, n3 = 0
flft»(l+ Ja)(l+ J3)- ^
u(l+ 4) = 0
a B cjiy^ae n
t= 0, n
2= 1, n
3= 0
< ^ ( i+ d
1) ( i+ 4
3) - e f
3 a( i+ 2 i
2) = o
o6pa3OM, T f t At- P .+ JO' " 20+ A) '
/
- *
.zjaeT BO3MO>KHOCTB BxoAHiuHe B (3.9) H (3.10) BenHtiHHBi TU 3 T2 2 Bbipa3H Tb
B 3KcnepHMeHTe P, q, At,A2.
H anncaH a n o iwaTepnanaM Równana, 3a»iHTaHHoro Ha i y IIOJIBCKOH KOH(|)e-„ 06o jio ^ e^ H Ł ie KOHcrpyKi^HH, TeopHH H n pH n oM eH H a", X I 1986 i\3
, IIH P.
JlH Tepaxypa
1. X. M . MyiuTAPH, HeKomopbie o6o6tą enun meopuu IHOHKUX u OÓOJIOHBK C npunootceHunMU K 3adaue ycmouHUBocmn ynpyzozo pasnoeecuH, H3BecTHH <J)H3. - MaT. o6mecTBa n p u K T y , T . X I , c e p . 3, c r p .
71 - 150, K a sa m , 1938.
2. X . M . MyuiTAPH, Hexomopue uOothą euun meopuu mouKux o5ojioneK, I I M M j T. I I3 Na A, c i p
. 439 -456, 1939.
3. X. M . MyuiTAPHj O6 oGnacmu npujnenuMOCTtiu npuójiuoiceHHOu meopuu oBoAonex. .H AH 'C C C P , T . 58, 1947.
4. X . M . MyiiiTAPH, O6 o6/ iacmu npujuenuMOcmu npuOjiunceHHoii meopuu O6OAOHBK Kupxzo$a- JInaa. I I M M , T . XI , Ms 5, 1947. 5. X. M . MyuiTAPHj O6 onpede/ ieiiuu de^opjuauuU cpedtamou noeepxuocmu OÓOJIOHKU npu npoiasoAhuux U3iu6ax. T p . K XT H , N° 13, 1948, Kasam.. 6. X. M . MYIIITAPHJ KanecriieeHHoe ucc/ iedoeahue nanpnoiceHHozo cocmomaiH ynpyaou OSOAOHKU npu MOJUIX detfiopMatfUJtx u npou3eoAmux cMeuteuunx. n M M , T . XI I I , JN° 2, 1949. 7. K . 3 . TAJIKM OB, YpaewHUH paemeecuH meopuu ynpyiocmu npu Koueunbix nepeMeufeHUfix u ux npu-AOMcemin K meopuu ynpywcmu. VISB. Ka3aHCKoro (JmnHana A H C C C P , c e p . (J>H3. - MaT. H ayK, Na I , 1948.
554 H. T. 8. K. 3 . FAJIHMOB, Odiifa/i meopuu ynpymx o6oAO<teK npu KOHBUHUX nepeueiaeuunx. TaM JKC, Na 2, 1950. 9. K . 3 . FAJIHMOB, K oBufeii meopuu wtacmun u O6OAOHBK npu KouenHbix nepeMeiaeuunx u de<fiopMauuxx. n M M: T. XY, Kg 6, 1951. 10. K. 3 . FAJIHMOB, K eapuauuoHHUM MemodaM peuieuun 3adau HenuneuHou meopuu nnacmuH u oSojioieK. • H3B. Ka3aHCKoro (pHtrijiajia A H C C C P , c e p . 4>H3.MaT. H rexn. HayK, N° 10, 1956. 1 1 . X . M . MyuiTAPH, K . 3 . FAJIHMOB, HemmuHaa meopun ynpyzux o6oAoueK. TaTKHHroH3flaT, 1957. 12. K . 3 . FAJIHMOB, P . F . C y n o m , O pacomax KCUUHCKUX yneuux no meopuu nnacmun u o6oAoneK. C6. „Hccn. no TeopHH njiacrint vs. oSonoieK", N° 5, H3#. KW, Ka3aHb, 1967. 13. K. 3 . FAJIHMOBJ O neKomopux uanpaeneHunx pcaeumux MexamiKU deffiopMUpyeMOZo me/ta e Kasauu. C6. „Hccjie^oBanHH n o TeopHH mracTHH H OSOJIOHCK", JVS 9, Btefl. KTY, Ka3am., 1972. 14. K . 3 . FAJIHMOB, O ueKomopux HanpasMuunx paieumux jnexamotu de^opMupyeuozo meepdoio me/ia e Ka3anu. C 6 . 3,Hccjiefl0BaHHH n o TeopHH iraacTHH u o6oJio^eK", Bbm. 14, 19793 cup. I I 82.
15. M . C . KOPHKUIHHJ HemmtuHue sadauu meopuu tiAacmwi u nojiomx oSoAonex u Aiemodu ux pewemia. H3fl. „ H a y K a " , M . , 1964.
16. M . C . KOPHHUIHHJ <t>. C . HCAHBAEBAJ FuOKiie ti/iacmuHbi u naue/iu. M . „HayKa'% 1968.
17. H . F . TEPEryjiOB, M3iu6 u ycmottnueocnib momux nnacmuH u O6O/IOIBK npu nomynecmu. M . „ H a y K a " , 1969.
18. M . A . HjitrAMoB, KojieGamtn ynpyzux o6ojioueK, codepoicataux oicudxocmb u za.3. M . 33HayKa",
1969.
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S t r e s z c z e n i e
ROZWÓJ N IELIN IOWEJ M ECH AN IKI POWŁOK W PRACACH SZKOŁY KAZ AN SKIEJ Praca zawiera szeroki przeglą d prac i monografii kazań skiej szkoł y dotyczą cych liniowej i nieliniowej mechaniki pł yt i powł ok. Podano w niej klasyfikację zagadnień ze wzglę du na charakter deformacji oraz geometrię powł oki. Omówiono również wyniki otrzymane dla zagadnień powł ok anizotropowych, termo-sprę ż ystych, plastycznych oraz z materiał ów kompozytowych.
S u m m a r y
D EVELOPMEN T O F N ON LIN EAR MECH AN ICS O F SH ELLS I N TH E KAZAN SCH OOL The paper presents a wide review of Kazan school publications and monographs on the linear and nonlinear mechanics of plates and shells. The classification of the problems with respect to the geometry and deformation of shells is given. The results obtained for the problem of anisotropic, thermoelastic, plastic as well as composite shells are discussed.