development of nonlinear mechanics of shells in the kazan school

development of nonlinear mechanics of shells in the kazan school

M E C H AN I K A
TEORETYCZN A
I STOSOWAN A
4, 25, (1987)
PA3BHTHE HEJIHHEHHOft MEXAHHKH OEOJIO*IEK
B TPyflAX KA3AHCKOB IHKOJIBI
H . F . T E P E ryjiO B .:• .• •:'
Ką sam, CCCP
1. Ilpocfieccopy. X. M . M yurrapH ( 1900- 1981) H npotj)eccopy K. 3 .
(1910 - 1986) Bbinajia qecTB ocHOBani- iH Ka3aH CKoń uiKOJibi H ejiH ueH noft MexaHHKH 060jionei- c, M iiorne cJjyHflaMeHTajiŁHbie pe3yjiBTaTti KOTopoii B STOH o6jiacTH 3iiaHHH
B 3OJIOTOH 4ioHfl; MHpoBOH HayKH. ITepBon TaKoro pofla pa6oToń (1938) SbiJia
f(HCcepTar(HH X . M . M yirrcapH ,,H eKOTOpbie O6O6W,CHHSI xeopH H TOH KH X
c npHHOH<eHHHMH K 3aAaie ycToft^H BocTH yn p yr o r o paBH OBecH H " [ 1 , 2 ] , B KOTOP OH
Bn epBwe B M H P OBOH H H TepaType 6&IJIH npoanaJiH 3H poBaH Bi cooTHOiueHHH TeopH H 060jio^eK flJiH cjryiiaH n p o r n 6 o B , cpaBHHMbix c TOJIIH H H OH oSonotiKH , H 6Ł I J I H BbinH caH bi
KOppeKTHbie COOTHOIIieHHH 3TOH TeOpHH flJIH OpTOTpOIIHOrO H H 30TponH OrO JIHHeHHO
yn p yr o r o Tejia. 3T a n n oH epcK an pa6oTa Bbi3Bajia IIOTOK HccJieHOBaiiHH B STOM H an p aB. X . M . M yuiT apH B CBOHK ny6jiH Kai;H 5K [3 - 6] flaeT aH anH 3 n p e ^ e jio B n p n ~
npH6jiH>KeHHŁix TeopH H , ocHOBaHHbix Ha rH n oTe3ax KH pxrocJja- JlH Ba ( 1947) ,
BbiBOfl TO^Hbix BBipa>KeHHit flJiH KOHe^Hbix 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 KpaeBoro atJx^eKTa. K i n c ji y (J)yHflaMeHTajibHbix H ccn ecjieflyer c^irraTB ny6nHKaD(Hio X . M . M yn r r a p H ^Ka^tecTBeH H oe H CCJie^oH anpH H ceH H oro COCTOHHHH 3TipyroH OSOJI O^KH n p H M a n t ix pctyopniaujzHX. H n p o H3BOJiBHbrx CMemeH H H x" [ 6] , B KOTopoii Bn epBbie 6bM a flana oSmienpHHHTaH T en epb
TeopHH o6ojio^eK Ha cjia6biH , cpeAH H ił H CH U BH BIH H 3rH 6Bi H KaaHajiH3 KpaeBoro stjupeicra.
B Tpyflax K . 3 . FanHMOBa B STH TOABI H H TCH CH BH O pa3BHBaiOTCH M C T O ^ M H ccicefloo6ojio^eK n p H KOHeqHLrx nepememeH H H x H M anbix fle^opMaipKH X B qacTH n o c r caMHX ypaBHeHHft [7 - 9] H BapHartHOHHBie (J)opMyiiHpoBKH 3ap,a.n TeopH H 060[10 H AP- ]v 3 T H H M irorne 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 SBIJIH 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 o6o6m eH SoraTBin OIIBIT Ka3aHCKOH U I KOJI 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 mexaHHKe
6wjiH n on ytieH bi yqeHHKaMH X . M . M yu r r a p H H K . 3 . FajiHMOBa — npocpeccopoiw.
542 M. T.
M . C , Ca^enKOBbiM A. B. H ApyrHMH, CCBIJIKH Ha KOTopbie MO>KHO
B o 63o p a x [12 - 14]. 3 T H pe3yjib- raTbi Sbijin noflbiTOHceHbi B KHHrax M . C . K o p [15], M . C . KopH H iiiH H a H <£>. C . H caH SaeBoft [16], H . F . TeperyjiOBa [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 pcK oro [ 20] .
H I H P O K O pe3ynbTaTbi yye H t ix Ka3aHCKOH uiKOJibi nejniHeHHOH MexaiiHKH o6ojioyei<
B peryjiH pH o H3flaBaeMbix „ H ccjieflOBarom x n o Teopn ii njiacrH H H OSOJIOn p n Ka3aHCKom rocyAapcrBeH H OM ynH BepcH TeTe (BbinycKH c I n o XYU c 1962
no 1985 r.r.).
B KpaTKOM CO06meHHH HCf HMKEKOH B03JV10H<HOCTH OCBCTHTb BCe BOnpOCbl, KOTOpbie
6biJiH n oflBeprn yTbi H ccjienpBaH H io B Tpy/ jax Ka3aHCi<HX yyeH bix. E C JI H OTMCTHTB OCH OBH bie H anpaBJieH H H H ccneflOBaH H H , TO K HHM a i e ^ ye r OTHCCTH cjieAyiomH e.
O S Bo n p o c bi HenHHeHHoM Teopiffl yn pyrocTH n o nocTpoeH H io onpeflextH iomiix
H o6ni;He BapHau;HOHHbie npHHU,Hnbi 6bijiH H ccjieflOBaiibi B paSoTax K . 3 .
[21 H flp.] H H . F . TeperyjiOBa [22, 23 H p.p.]. B ny6jiHKau(Hn [24] ^aeTCH
o 6o 6m eiiH e cjjopmyji (1.1) p,nx O6OJI O I I KH i<ai< TpexM epH oro Ten a.
Bo n p o c b i nocTpoeH H H yTOiH eH H bix Teopnft miacTH H H o6ojiotiei< 6biHH n oflBepn iyTbi
nccjieAOBaH H io B paSoTax X . M . M yixiTapn H H . P . TeperyjiOBa [24, 25] , A. K . F a jiH H bina, A. B . CayeiiKOBa c y*ieHHKaMH H p,p. B CTaTte [23] TeperyxrosbiM H . F . Bn ep B t i e AaH a H enH H eiiH aH (J»opMyJiHPOBKa o Sm ero BapHan,HOHHoro npH H i^nna, 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 apsi [ 26] , M . A . H n traM O Ba, H . K, FajiHivtoBa, A. F . TeperyjiOBa H ji,p.
^ n c jie iiH bie MeTOflbi pemeH H H reoM eTpH ^ecKH HejiHHeiiHbrx 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 3a ^ a u noji3yM:ecTH H 3yiiajiH ci. H . F . TeperyjioBbiM , M . C .
IIlHpOKHH K pyr 3aAa^I O COCpeflOTO^eHHblX BO3flei'lCTBHHX H KOHTaKTHOM B3aHMOAencTBH H H ccjieflOBan B pa6oTax K ) . I I . JKnrajiKo, H . F . TyptaH OBa, K ) . I I , ApTioXHHa.
A. B . Ca^enKOBBiM 6WJ I pa3BHT yfla^H biii MeTOfl pem en H H 3aflaŁ i CTaTHKH H flH H aM H KH o6ojioM eK, noJiy^H BU iH ft Ha3BaHHe TeoperHKO- 3KcnepHMeHTajiLHoro M ero^a, yc n e u m o pa3BH TBiit H npH M eH eiiH biii K niHpoKoiwy Kpyry npH Kna^H bix safla'q K ) . F . K o H on jiesbiM H flpyrH MH e r o y^ieHHKaMH.
B MOHorpadpHH K . 3 . FanHMOBa [27] noABefleH H Tor e r o 6OJIBU IOH paooTbi n o n o c r p o eH H io HexiHHeHHOH MexaHHKH JiHHeHHO ynpyrH X oSonotteK 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 un en K. 3 . FanHMOBa rpyn n o ft aBxopoB H3Jio>KeHbi
p ^ A pe3yjiBTaTOB n o TeopH H H npaKTHKe 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 yr a H H a [29] H3JiaraeTCH pa3BH Toe
H M H H anpaBJieH 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
[ 12, 14] , H ynoM H H yrbix Bbim e C 6OP H H KOB „H ccjieAOBaH H H n o TeopHH njiacTHH H 060Jio^iei< ". H axoflH TCJi B n e^aTH KHHra K . 3 . F anH iwosa, B. H . riaSm yiiraH a, H . F . T e p e r yji o sa ,,OcH OBaH i«c HeJiHHeiiHOH TeopH H oSojioqeK 5 ', B KOTopoft
MEXAHHKA OBOJIOMEK B I<A3AHE 543
H 4>H3iraecKHe cooTHOiueHHH AiexaHHKH o6oJio^ieK [ 30] . H . C . F aH H eB n ccjieflyer n e jiHHefiHbie safla^H H 3ra6a o6o^o^eK c yuexoM nuacTH H ecKH x fletJiopMauH ftj a C . B . ^ e peBam<HH c y^eHHKaMH H 3y^aeT n ec ym yio cn ocoSiiocT t OSOJIOH KH H 3 BOJIOKH H CTBIX
KOMnO3H TOB.
2. O co 6o cneflyex OTMCTHTŁ oflHy M3 H an 6ojiee flpraix ny6jiHKaŁ(HH X . M . M yu iT apH
- ,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, npeAcraBJiH iomH X o c o 6biit H H Tepec. OrnpaBJiH H CB o x ypaBH eH H ił paBHOBecHH
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 6e3pa3iwepHbiM nyTeM yMHOHceHHH Ha BeJiH ^H H bi man E mL n
— cooTBeTCTByioiuHe C Teneim , E • — MOflym. K ) H r a , L — xapaKTepH bift n o n e pa3M ep OSOJI O^KH H JIH n jiacxH H ti, X . M . M yu iT a p n n o ^ Be p r a e x ypaBH eH H H
TeopiiH yn p yr u x oSon o^eK KaiecTseH H OM y aH anH 3y. B Ka^iecrBe ocH OBH oro
n apaM eTpa, 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)opHa n p e^ ejie nponopu;H OH ajibH ocTH . BBefleH ne STOFO n apaM expa B K aiec T Be 6a30Boro onpaBAaH o xeM, *ITO B peajibH bix ycjioBHHX npuxoflnTCH HMCTŁ ASJIO 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 bie H arpy3KH , yp o K o xo pt ix onpeflextnexcH flonycnaeMbiMH Be^HMHHaMH HanpjDKeHHH H J I H
., nocjieflH H e H3 K o xo psix HMeiox n o p n ^ o K sp. I I on biTKa on paB^aTb
T eopm o oQojioieK npraiH TH eM npeAnojio>KeH H H o XOM, mo „SyA^M pacoviaTpH Baxb 6e c Majibie nepeiYiemeH H Ji", H e
Bbi^epwcH BaeT K P H T H K H , xaK KBK 3TO MO>Ker n p ii-
K o^eH b HH3KOMy ypoBH io flonyci<aeMbix K paccMOTpeHHK) Ha ocHOBe TaKHX yp a BH arpy3OK H HanpH>KeHHO- fl;ec})opmHpoBaHHoe COCTOH H 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)opMH poBaH H oe COCXOH H H C,
TO ecrb onepauH H flH c^dpepeH tiH poBaH H H He M eH ser nopnflKa 43yHKi(HH. J ^ p yr o t i
HanpH>KeHHoro COCTOHHHH TaKOB,
544 H . r .
rp;e X — noKa3aTejib H3M.eHHeM0CTH Hanpa>KeHHoro COCTOHHHJI. B npoijecce
KaiecTBeH H oro aHauKraa cyH TaeTca, TTO
2Fh
7 "*" = l E h A s ^ , M =
~
3
%
V
,
H JIH n ocjie o6e3pa3MepHBaHHff
T " ~ hAafieye0Y, M* ~ A M ° * % r ,
13
al>
2
ryje 2h — oxH eceH H aH K L Tojim n H a, T * H M — OTH eceH H tie K EL H EL ycvuw.fi
H MOMeHTbi, x ap — yMHOHceHHEie n a L napajwexpbi H3MeireHHH K P H B H 3I I , 3iiar< ~ yi<a3bm a e x H a offHi- iaKOBOCTB n opa^K O B conocraBJiH eM Bix BC JIH ^IKH . I<jiaccH(|)HKaB[HH oCojio^eK n o reoM eTproi n pefljioiKen a .
cjieflyioiqaH :
I< pH BH 3H bI,
p ]p KpBH 3H BI ,
4
K/ } ~ b ~ £ p' — noJiorne OSOHO^IKH,
ioc/J ~ b ~ 1 OSOJIOIIKH KOHe^IHOit K pH BH 3H bI .
XapaKTep Ae4)opMHpOBaHiioro COCTOHHHH X. M . M yuiTapn KJiaccH({)nqHpyeT
KHIJHM o6pa30M (x xp ~ %, oia ~ co): •
« ~ 1 CHJIBHblft
H 3 r H 6 e H MeflJieH H O H3MeHHK>meMCH H anpH >KeH H OM COCTOHHHH
^ B
, a
,n,jra oSojio^eK KOH CIH OH KP H BH 3H BI b ~ l(THn la ) c norpeimiocTbio
E p
B c p a BH e H H H c eflH H H ijeft n pH xoflH M K o 6m eH 3Bec T H Ł iH ypaBH eH H H M JiHHeÓHOH
O 6e3M 0M eH TH OM H anpK>KeH H OM COCTOHHHH
V a r a ? + / = 0 , bipT^ + g^ 0. (2.5)
,E[JIH n u a C T H H M ajIOń KpH BH 3H bI b <, S p (THTI 1 B ) yCJIOBHH COBiWeCTHOCTH npH BOflH TCH
<**(*«?*?fietu = 0, c^V«^, , = 0, (2.6)
(2.7)
Kai< ypaBHeHHHM ( 2. 5) n p n ^ = 0 MTOKHO C norpeuiHOCTBK) e p y^oB^eTBOpH TB,
n o jia r a H
«t) = cw- c^v^v, *. T
(2.8)
B m o r e ypasH eH H ii, peuiarom ae s a ^ a ^ , npiiMyx
7 y_ • v V 0 = O A/ / (2 91
• •
MEXAHHKA OBOJIOMEK B KA3AHE Afft%l 545
- <T2) = aaapa"r + (1 - cr) cfa0*,
a — KoacJxJfflUHeHT I lyacco H a.
Jinx o6oJioqi<H ManoH KpHBH3Hbi b ~ \ / sp ( r a n 1 c) ycJioBH H coBMecTHocTH c r r o r p e m HOCTblO j/ fip H pH M yT BH fl
c
^ V ( V a V^ B w + bev Map) = 0 , {My *a? = 0 .
3flecb on aTb BO3MO>KHO npeflCTaBJieH ne ( 2.7) , a n epBbie flBa ypaBHeHHH paBHOBecHH
( 2. 5i) n pH gf = 0 yAOBJieTBopHiOTCH nocTaH OBKoń ( 2. 8) . TaKHM o6pa3OM, B STOM cjiy• qae umeeiw cHCTewy flByx HejiHHeHHbix ypaBHeHHH BH ^a ( 2.9) , ( 2. 10) .
H 3rn 6e H ~ | / e p H n pw 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
K
Va r " + ^ = 0, (ban + ^Tf + q = 0,
H AonycnaioT jiHHeapH3ai^Hio B nocjieflH eM ypaBHeHHH JiHiiib c n orpeuiH ocT bio jiHHeapH3yiOTCH c norpeniH ocTBK) sp. p,o
]/ sp,
r«e:
JI.JIH 06oJI0'tIKH M3JIOH KpHBH3HbI (Z) ;$ \ / Sp) (THII 2 B ) C nOrpeiUHOCTbK) S p B CpaBHeHHH
c eflHHHi^eft ycJioBHH COBMCCTHOCTH Ko^am ^H ( 2. 6 2 ) n n e p B t i e J\ B& ypaBH eH H H ( 2. 5 a )
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 jiHHeapH30BaTb aa>Ke n p e H e 6p e r a n
OCOSCH H O HHTepecHbi BbiBOAbi n o Ka^ecTBeHHOMy aHajiH3y K paeBoro
I O K oi<a3ajiocb3 HejiHHeHHocTH B p a ge cjiy^aeB H rpaioT cymecTBeH H yio
I lycTb JIH H H H Xi = c o n st coBnaAaeT c JIH H H CM r p a H n n w cpeflHHHoft n oBepxn ocT H
H H e KacaeTCH acHMirroTH*iecKOH JIH H H H STOH n osepxH ocTH H
$'> "Ó] m > 1, n ^ O, s > 1/ 4.
3T0M
£ a/ 3 ~ telx e, HJIH h ~ fi™~"~'.
546 H . T .
H 3 ypaBH eH H H H 3ra6a (2.2) c u e ^ ye r , U TO
/? = m- n- 2s—t,
T o r ^ a i<aK H3 ycnoBH H COBIACCTHOCTH RecJropMauHH (2.4) (ycJiOBHe F aycca) c jiea yc r ,
£ m m- n- 2s.
ConocTaBJiH 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 oro n o p H / n o c H 3rn6H biMH . H 3 n e p Bt ix 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
2 s
Tn ~ r 1 3 4 ~ T22s p ,
6) AJ1H
o6pa30M , n p n s ^ 1/ 4
KOHCTHOH KpHBH3HbI b ~ 1 (/? = 0) n pH fia/ J ~ e p (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> He 6ojiee yep B ypaBH eH H ax MO>KHO H OJIOJKH TB
^22 + ^22 — ^22 H Te
M CaMfclM. JIHHeapH3OBaTb K ypaBHeHHH B yCHJIHHX H 3e(J)0pMat(HSX.
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 ep e3 nepeMemeH H H ^OJI>KH O 6BITB coxpaH eH O B
H OM
B cjiytiae 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 3rn 6a o ie ^ ye T n n c a T t B HejiHHeftHOM BH ^e ( 2. 2) , TaK Kan
la
nopH flKa sp B cpaBHeHHH c eflMKHqeft BHeceT Sojit 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 emiH BO3MO5KHa c n o rpeu iH o cT t io | / e p . OTM C TH M ,
qxo JiHHeapH3aE(HH BŁ ipan<eHHH e l t Mepe3 nepeMemeH H H B03M0H<Ha JIH U IB C n o r p e u i 4
HOCTBK) fii' .
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 paMeTpa 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 paeBo ro
3(|)4)eKTa. O npeH M ymecTBax n epefl 3TH M H KJiaccH(|)HKai|HHMH BBCACH H H KaK ocH OBH oro
Jioro rrapaMeTpa BejiH<- iHHŁ>i eap ~ sp ci<a3aHO Bbiiue.
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 cnpeH H oro X . M . M y m i a p n , H npeflJiOMceno CM H TaTB
2
co ^ O ( 0 ) — M anoe
to = 0 ( 0 ) — cpefln ee
to — 0 (\ / &) — SoJttuioe
to = 0(1) — Kon e^H oe
2
Jm<aił H H n p « & ~ ep npeflCTaBJiH ercH KoppeKTH oń H BBefleH ae 6ojn .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 TeopHH n o c T p o e m o i
cooTHomeHHH flJiH aH H 3OTponH bix oSoJio^eK, B qacTHOCTH, fljiH KOMno3HTHŁix, o 6p a 30BaH H bix H3 MOHocnoeB oflHOHanpaBJieHHO apMH poBaH H oii crpyK T ypbi [ 22] . 3 T H M H
HCCjieflOBaHHHMH BOcnojiHHeTCH n p o 6eit B HejiHHeHHoft iwexaHHKe TOH KH X oCojio^ieK
B nacTH nocrpoeH H H o6m eH TeopH H A- ns o n p e ^ e n H io m n x (4)H 3H iecKH x) COOTH OH ICH H H ,
T o r^a KaK Bo n p o c t i TeopHH reoM eTpH qecKoft HejiHHeiiHOCTH H a cero^H H ^ocTaToiH O
rnySoKO pa3pa6oTaH bi. B ^ a c r n pa3BHTHH HccjieAOBaHHH HejiHHeHHbix 3afl;a^, cBH 3aH H t ix c KOHeHHbiMH BpauieHHHMH, cneflyeT OTMeTHTb ycn exH IIojiBCKOH uiKOJibi M exaH H KH o6ojioqeK BO rjiaBe c npo(J). B . n e r p a u iK e BH ie M .
a. H a n oBepxH ocrH npH BefleH H a S o B He# ecpopMHpoBaHHoivi COCTOH H H H BBefteM CH CTejwy jiarpaH H <eBbix Koop^H H ax ^ "( a = 1, 2) c MerpEreecKHM TeH3opoM a^ = (pa' Pp)
H KOOpflHHaTHblMH BeKTOpaMH pa — dp/ dx*. H a 3KBHflHCTaHTHŁIX n o HOpMaJIBHOił K S o
3
Z = X nOBepXHOCTHX S z C OpTOM HOpM3JlH m KOOpflHHaTBI X°* HMOOT MeXa
TeH 3op gap = ( r a • rp), r = p+zm, H KOopAHHaTHbie [BeKTopbr r a = 8r/ dx
I I p H 3TOM g33 = 1, ga3 = 0. B fleCJJOpMH pOBaH H OM COCTOHHHH Ha S$ HMeeM a*p =
= (P *"P *). pS = de*/ dxa, 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
OTHeceHa K H a^aJitH OMy COCTOSJHHIO H o6pa3yeT TeH 3op
T e H 3 o p HanpHH<eHHii K O H I H Z 1 HMeeT cocraBJiH iomH e a ifc B ^e^opM npoBaH H OM 6a-
a TeH 3op yaioBH Lix HanpHHteHHH T uuvieer B He/i;ecpopMHpoBaHHOM 6a3H ce r< cocraBJiH ik
lk
t H T = t rtrk fc
tk
n p H BBinojiHeHHH ycjioBHft / ' = cr / c*, » rfle g^ ; i—iyioTH OcTb
B 3e4)opMHpoBaHHOM cocTOHHHH. B cTaTŁe [22] noKa3aH o 3 • mT
O B cjiy^ae rtojiH o ro pacceH H n a: B Bnp,e Tenira p a So r bi BHyTpeHHHX HanpH>KeHHH Ha npH pameH H H x RC)
n on 3yqecTH de\ f? H njiacTH ^ecKH x fle(j)opMar(H H de\ l HMeeT wecTO 06064>opMyji
delk
5eik — deik — de\ ]? — ds^, eik —noxtH we flec^opMai^H H , F-
detk — n pn pam eH H H yn p yr t i x .gecpopMainiH . 3flecB 3HaK <5(...) yKa3biBaeT Ha T O ,
n pH paiueH H e cooTBeTCTByromeH BejiH ^H H bl He e c n . ITOJIHBIH fl;n4)4)epeH i;H aJi, a n ojiH oe
n pupam eH Bte cooTBercTByioineH BenH iH H bi 3aBHCHT OT nyTH , n o KoiopoM y npoTeKaeT
n pouecc.
( 3. 1) , o C o m aio iu an cpopMyuy F pH H a H a cnyqafl H an n qaH H e yn p yr a x n p n fiopMainiH H cJieayeT H3 T o ro , ^ T O AJIH njioTHOCTH cBoSoflHoft 3H eprH H F
a cjiy^iae itajimma. TemiOBbix n p o n e c c o B HMeeM
ik
QtdF = a detk - Q*sdT- jJJ PJdXj,
:
- • • • •• • • !'. . t • • •. ;.• . • c i : Td - o n i . - v • • , dt+YM :
• • [ q ^
s — nnoTHOCTB 3H Tponraij T — aScomoTH aji y i r i i . ( 1 2 >
. . • . - ; , .. !• .:
TeM nepaTypa, r—H H TeH CH BH ocn>.
T en n a , q = g^ r f — BeKTop noTOKa Teim a, PJ — cocraBJiH ioiH H e o6o6iqeH H BIX CH JIJ %j — oSoSrą eH H we nepeM etneH H fl, H a KOTopux CH KBI P' npoH3BOflHT pa6oTy,
pacceH BaeAiyro B BH ^e xe n n a . B HccneflOBaHHH [34] n o K asan o , MTO He MeHee 9 0 % M exaH iraecKOH paG oTŁi n a njiacTH ^ecKH x «ec|)opManH H x nepexoflH T B TenjioByra
T a K KaK MexanH 3M fle<J)OpMaiiH H n pH rroji3ytiecTH ofltiaKos c MexaHH3MOM
K o r o fletJiopMH poBaH H H , TO H n pH nojrey^iecTH cjie^yeT o>KH,n;aTŁ n e p e xo ^ a paSoTbi
HanpH>KeHHH H a necJiopmaiCHHx noji3yiiecTH B Term o. TaKHM o6pa3OM,
ik
MexaH H qecKaa: a n e p r n H a de\ ^ T o r fla corjiacH O (3.2)
ik
H SH eprn n a de$ n epexo^H T B Ten iro.
549
M E X A H H K A OBOJICfflBK B I<A3AHE
rpp npHHHTO
H 3 cK33anHoro cjie^yeT (1).
B. OyHKUHJi F 3aBHCHT OT HHBapHaHTOB fle^opMH poBaH H oro COCTOHHHH, TeAinepaTypw T H , mo>i<eT 6BITŁJ OT Apyrux napaMCTpoB npoijecca. B Ka^ecTBe
3e4)opMHpoBaHHoro COCTOHHHH Bbi6epeiw BenmiHHbi [22]
I 2 = eape^>
h = el
(3- 3)
/l(f)ga/? = 0, ya = 2ea3. KacaTenBHan K iSz njiocKOcrt ecTB nJiocnocTB
CBOHCTB. B 3TOM cjiyqae flJiH tm coraacH o (3.1) imeeM
(3.4)
3F
Qt* =
3«ecb ta = ta3, Ax — Q3Fj8I t. Rsin opTOTponHoro
H B ocax
OpTOTpOnHH A12 = 0.
Jinn TOHKHX oSojioieK t33 <4,tll,t22 H , cne^oBaTejiBHO, 8Flde33 ^ 0. TaKHM 06pa3OM, F OT e33 He 3aBHCHT. rt o n aran nonepeMHyio cABHroByio
(ya ~ 0)
B OCHX OpTOTpOIIH H flJIH JIOKaJIbH blX KOOpAH H aT C XapaKTepH C TH KaM H glx gl2 = 0 HMeeiw An = - A22 — g22 = U
= 1, A12 = 0
Ix**2elx, I 3 = 2e22, I 2 = 1e\ 2.
IlpHMeM, IITO CHcreMa HanpH>KeHHii / "^ paBHOBecHan, npHpameiiHH deal!
BO3MO>KHbie.
B CHuy Toro, I T O deaft — fle(J)opiviaqH H y n pyrn e npH gdtal1 — Aa^ oydee
• dtafdetlfi> 0.
- ex, e22 = e2, eiz = e3. M aKO3(J)4)Hu;HeHTOB Alk n opoK ^aer noitojKHTejibHO onpe^eJieHHyio
(J)opMy.
\ \ Alk \ \
E 2,
.
3
E x — ycJiOBHbie
yn pyrocra. l i p a 3TOM E t- HMeer
Ha
H . T.
.550
E lt E 2, E2
E3
, E 2 ~ E 2, E 3
- V2
M aTpH i;ti H ^ H flaer p, ik E t = JikiE k \ i,k = 1 , 2 , 3 (H e cyjwMHpo-
a yc ao BH e nojioH ttrrejibH oft onpeoeneH H OCTH KBaApaTH^Hon (|)opMbi £ dtk det dek
ik
— K ycJioBHHM E, > 0 , ftikfiti < 1; / , k - 1, 2 , 3 (H e cyiwMHpoBaTt) H an p H M ep , H3 yc n o BH H / tiia/ iai < 1, fJ, 12£i = ("21^2 cn cayeT ,
fl12 < ł ?
n o3o6H Bie o u e m a i fljia Bcex qjieHOB MaTpmi;Bi
noJiymHM,
3an n caT b cjie^yio m ee cooTHOuieHHe
1,
r\ , rf,
x2r]2
o6pa3OM , ecriH B, = QdF/ deit T O
ż'
8B2
n p n ?; - >• 0 HMeeM, ^ T O cymecrBeH H BiM 6yfler JIH IIIB OAH H
BB1ldel~E 1.
(3.5)
x - > 0 ^aeT cymecrBeH H Bie
( 3. 5) cooTBercTByer H H TH H OH MOflejiH o^HOHanpaBJieHHo apMH 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
OC H Qxi.; E t a 4600 M i l a , ^ 2 ~ 18000 M i l a , G12 r]2 ~ 1/ 3, x2 ~ 1/ 3, <X 4500 M i l a H , cneflOBaTenbHO,
7] ~ x.
X O T H 1/ 3 H H e Mano B cpaBHeHHH c e^H H H ueiij oflHaKO B ciu iy acH M irroTH ^ecKoro
xofla n o Majiojwy n a p a M er p y npH BefleH H Bie orł eHKH KOppeKTHBi.
JXns rj ~ x H3 ( 3) cue^yeTj MTO dB^jde^ B acHMirroTirqecKOM pa3JioH<eHHH
HaerrcH c H Jien oB, co^ep>Kait(HX rj B H y^eBoft cren eH H , dBi/ de2 n ep BBie cren eH H rj, 8Bxj8ez — c qjieH
— c qjieHOB, coAep>Kaiu(HX KBaflpaTBi rj.
2(.ei> e2, e3,rj)
MEXAHHKA OEOJIOIEK B KA3AHE 551
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
B2 = W i O O + ^ Va fc i. e2) + rf^ 3{ex,e2,e3,r])
ei,e2., e3,rj)
B acHMiiTOTiraecKOM pa3Jio>KeHHH
= y) 3k = ^ 4 k = 0 npn k > 1. H 3
c yq e r o M T o r o , ^ T O n p w ei2 flecJ>opMaitH H e n , e 2 2 H e At o r yr Bbi3BaT b H a n p H ł K e H H H t i 2 ,
0*22 = - #21 ( e n . ^22)^11 + - ^22(^11,
E C JI H B acHMirroTirqecKHX pa3JioH<eHHHx jjjin JSX H B 2 orpaHHqHTBCH
, TO onpeflejiH iomH e cooTH om em ra flOJi>KH bi 6BITB npeflCTaBJieH bi B
e^ii = S 11(en)en+£ l2(e11)e22,
t = JB2i( eii) e
Q 22
B npocreH iueivi cjiy^ae, K o r^ a B pa3Jio>KeHHH AJI H BX coxpaH H ercH O,II;HH qjieH , TO ecT 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 [ 35] .
c . Oim uieM n p o it e ^ yp y 3KcnepH MeH TajiBH oro on pe^e^eH H H cjyyHKą im
At(Ij)
orpaH iweH H H X y a ~ 0 , de\ p = 0 , deffi = 0 , T .e. B c jiyqae Majibix n o n e p e ^ H b ix
ro B. Jlflu ycH jim ł , OTHeceHHbix K HaqanbHOMy HeflecpopMHpoBaHHOMy COCTOH H H IO H B
Hefle<J>opMHpoBaHHOH
f - h f - h h
, J
T12= Jet12dz, (3.7)
- hh
rp,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 l e 9 * —
yrjibi, KOTOpbie o6pa3yioT H anpaB^eH H H apMH poaaH H H c o6pa3yK> meii
552 H . T. TEPETYJIOB
KOJiH^ecTBe cjioeB B O5OKX HanpaBJieHHHX. r i p n OTOM flJis cjiy^iaH e12 = 0 H3
(3.3)
1
z
I 3 = 2(ell
21? = (en~e22) 2sm22(p,
If = 2(eilcos <p + e22s'm <p), A\ x = ~ A\ 2 = c o s 2 ^ , A\ 2 - si
Tai< i<aK I j OT HOMepa CJIOH He aaBHCOT, TO coraacH O ( 3. 4) , (3.7)
T tl (en- e22)siii22f,
= 2h[A,_ • ( l+ c o s2 < p ) + ^ 3 • (1- cos295)]+ 2hA 2 - T 22 2
=2h[A1(l- cos2<p)+A3(l+cos2<py]- 2hA2(e11- e2Z)sin 2<p. Jlna on pe«eJieiiH H A2(l2) ®®
OTCIOAS n p n cp = ?r/ 4 nojiy^H M (J)opMyjiy
4hA2(e22~e11) = T Z2- T XX (3.9)
B n p eAejiax TO^H OCTH pacraioTpeH H oro paH ee acH M rrroTimecKoro rrpH6jin>KeHHa A2
3aBHCHT JIHIHB OT I 2 > flJIH KOTOpOrO n pH fp = jr/ 4 HMeeM
(e11- e22) 2.
2/ 2 = I I p H T 1X—T 22 > 0 H 3 (3.9) AJIH A2 nojiy^H M
A2(I2) = ^ i ^ Ł .
4/ JJ/ 2/ 2
B npe,n;ejrax TOH »ce acHMirroTHiiecKOH TOI H OC TH A X = A^Ix,!^, A3 = Az{Ix,I 3).
onpe^eJieH U H STH X ^ymax,ii& H3 (3.8) n p a cp ^ njA H H 3BCCTH OH ć pyKKaym A2(Iz)
= T ncos2(p—T 22sm2(p—2hA2(eXi. 2
4hA3cos <p i
— e
2
2
2
) q ,
2
= - T 11sin <p+T 22cos <p+2hA2(.en- e22)sm 2<p. '
3 fle c t Bee BxoAH mH e H BH O e n , e22, h cJieflyeT Btipa3H TB Tiepe3 I t H 7 3 coraacH O ( 3. 3) ,
TO ec r b
26^00521)? = Iy cos, 2q — 7 3 sin 2 < p,
2e 2 2 c o s2c > = 7 3 c o s 2 c > —Xt sin 2 ^ ,
^cp
At, A2,
CBH3B MOKfly 3aMepHeMBIMH B 3KCHepHMeHTe
A tki- ds,. , ds~ ds2
H BeJIHUHHaMH e u , ^ 2 2 , KOTOpaH flaeTCH B
C flpyroft CTopoH bi, ecjiH K i;njiH H flpy npHJio>KeHa o ceBan pacTHrHBaK>maH cnjia P H BHyTpeH H ee flaBjieH H e q, TO n o n H a n oceBaa cHJia n p H 3aKpBiTfcix flH mn;ax Q = PA- nR^q
CO3flaeT B I^HJIHHflpe HCTHHHBie HanpH>KeHHH
^ £,a Q Stu Q
&
* • " • ' ' " I ' 1 • ' • ' • • • • '• < ''• • '• '
MEXAHHKA OBOJIOIBK B KA3AH E 3flect i?* = R(l+d2), 553
K = h(l+A3). C yqeTOM nepexo^a
H3 BapH aijuoH H oro ypaBHeHHH n pH 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 *. Orcio;ja c n e^ yer , MTO B cjryyae «j = 1 ,n2 = 0, n 3 = 0
flft»(l+ Ja)(l+ J3)- ^ u(l+ 4) = 0
a B cjiy^ae nt = 0, n2 = 1, n3 = 0
< ^ ( i+ d 1 ) ( i+ 4 3 ) - e f 3 a ( i+ 2i 2 ) = o
o6pa3OM,
T f t At-
P
.
+ JO' " 20+ A) '
/
-*
.zjaeT BO3MO>KHOCTB BxoAHiuHe B (3.9) H (3.10) BenHtiHHBi T U 3 T 2 2 Bbipa3HTb
B 3KcnepHMeHTe P, q, At,A2.
H anncaH a n o iwaTepnanaM Równana, 3a»iHTaHHoro Ha i y IIOJIBCKOH KOH(|)e„ 06ojio^e^H Ł ie KOHcrpyKi^HH, TeopHH H npH noM eH H a", X I 1986 i\ 3
, IIHP.
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 I 3 Na A, c i p . 439 456, 1939.
3. X. M . MyuiTAPHj O6 oGnacmu npujnenuMOCTtiu npuójiuoiceHHOu meopuu oBoAonex. .H AH 'CCCP, 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 . TAJIKMOB, YpaewHUH paemeecuH meopuu ynpyiocmu npu Koueunbix nepeMeufeHUfix u ux npuAOMcemin K meopuu ynpywcmu. VISB. Ka3aHCKoro (JmnHana A H C C C P , c e p . (J>H3. - MaT. HayK, Na I ,
1948.
3 Mech. Teoret. i Stoi. 4/ 87
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, 1979 3 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 . 33 HayKa",
1969.
19. M . A . HJIWAMOBJ B . A. HBAHOB, B. B . FyjiHH., TJpoHHOcrm>s ycmounusocmb u dtmaMUKa 060jioteK c ynpyzuM lanoAHumeAeM. M . jjHayKa", 1977.
20. H . B . CBHPCKHHJ Memodu muna EyBnosa-Fa/iepKuna u nocAedoeawejibHux npuOAUoicemiii. M . „Hay­
K a " , 1968.
2 1 . K. 3 . FAJHIMOBJ K meopuu KOWMUX detfiopMatfuu. Yw. 3anHCKH K F Y , T. 109, KH. I, 1949, c r p .
3 5 ­ 7 1 . Ka3aH&.
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St reszczen ie
ROZWÓJ N IELIN IOWEJ M ECH AN IKI POWŁOK W PRACACH SZKOŁY KAZAN SKIEJ
Praca zawiera szeroki przeglą d prac i monografii kazań skie
j 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, termosprę ż ystych, plastycznych oraz z materiał ów kompozytowych.
Summary
D EVELOPMEN T OF N ON LIN EAR MECH AN ICS O F SHELLS 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.
Praca wpł ynę ł a do Redakcji dnia 5 stycznia 1987 roku.