HEJ ALAKVALTOZ AS UTANI KOZ EPFEL ULET ENEK MEGHATAROZ ASA ALAPFORMULAINAK (METRIKUS TENZORANAK ES GORB ULETI TENZORANAK) ISMERETEBEN Kozk Imre Mechniki Tnszek. Miskolc Pczelt Istvn 70. szuletesnpj tiszteletere. 009.
Egyuttmozgo koordintrendszer hej kozepfeluleten x 3 n= 3 x 3 =x 3 n= 3 x =x P ( S) r x Alkváltozás előtti középfelület x u r P (S) x = x ( S) en értelmezett mennyiségeket felülvonás jelöli κλ= κ. λ κλ= κ. λ Alkváltozás utáni középfelület
Nemlineris hejelmelet. Els}orend}u kozeltes (Kircho-Love hipotezis) Green-Lgrnge lkvltozsi tenzor E = e + k x 3 ; E 3 = e 3 = 0; E 33 = e 33 = 0 e = ( ); k = b b = e + ; b = k + b 3
Megoldsok r(x ; x ) meghtrozsr, h (x ; x ) es b (x ; x ) ismert W. PIETRASZKIEWICZ, M.L. SZWABOWICZ : Deternintion of the midsurfce of deformed shell from prescribed surfce strin nd bindings. Int. J. Solids nd Struct., 44, (007), 663-67: dierencilgeometrii gondoltmenet W. PIETRASZKIEWICZ, C. VALL EE : A method of shell theory in determintion of the surfce from components of two fundmentl form. J. of Appl. Mth. nd Mech., 87, (007), 603-65: kontinuummechniki gondoltmenet, sik kozepfelulet lekepezese gorbult feluletre W. PIETRASZKIEWICZ, M.L. SZWABOWICZ, C. VALLEE : Determintion of the midsurfce deformed shell from prescribed surfce strins nd bendings vi the polr decomposition. Int. J. Non-liner Mech., 43, (008), 579-587: kontinuummechniki gondoltmenet, gorbult (S) kozepfelulet lekepezese gorbult (S) kozepfeluletre 4
Kontinuumkinemetiki lposszefuggesek bf; bv; b T; b v: hrom D-s tenzor, vektor; T; v; T; v: ket D-s tenzor, vektor Alkvltozsi grdiens bf = k k = F+ 3 3 ; F= Az lkvltozsi grdiens polris felbonts bf = b R b U = b V b R; F=R U Forgstenzor br = b f k k = R+ 3 3 : b f k = b R k ; R = f : f 3 = 3 Jobboldli vonlelemrny tenzor (right stretch tensor) bu = b U T = b U l k k l = U+ 3 3 ; U = U T = U = U Az (S) kozepfelulet helyvektor r; = = F = R U ; kell: U es R 5
Az el}ods celj. Figyelem felhvs problemr, vlmint PIETRASZKIEWICZ es szerz}otrsi ltl dott megoldsokr. Megolds bemutts W. PIETRASZKIEWICZ, M.L. SZWABOWICZ, C. VALLEE gondoltmenete lpjn, elter}o formlizmus lklmzsvl Az el}ods gondoltmenete I. ; =)=) U, U b kl II. U b kl =)=) k R b l = U b kl III. U b kl, b ; b =)=) R, b IV. R, b R, b = b0 : integrlhtosgi feltetel, R, b =)=) R b : integrls V. R; U =)=) r; VI. r; r; = b0 : integrlhtosgi feltetel, r; =)=) r : integrls 6
I. ; =)=) U, b U kl Cyley-Hmilton tetel: b U 3 bu I b U + b UII b U b UIII b I = b0 h i bu l k = 4 U U 0 U U 0 0 0 3 5 ; U U = U U U bu = U + 3 3 ; b U 3 = b U b U = U 3 + 3 3 bu I = U I + ; b UII = U II + U I ; b UIII = U II U U I U + U II I = 0 =)=) U = U I U + U II I U = U U = U U =) U = U T U = F T R R T R = F T F = ( ) = =) 7
=) U U = = det = (det U ) det = (det U ) =)=) det U = p U II = det U = det U = det U =)=) U II = q U I = U = U = U ( + U II ) = I U + U II =)=) r I q U I = + U = r + q + q ; U b = U ; U3 b = 0; U33 b = 8
II. b U kl =)=) k b R l = b U kl br T b F = b U k b f k l l = bf k l k l = U b kl k l =)=) b f k l = U b kl br = b f k k bf = k b I k = b f k l l k = bf k l l k br = b U kl l k 9
III. b U kl, b ; b =)=) b R, : dierencilegyenlet z b R forgstenzorr br, b R T = b c b I; b R T b R, = b k b I br, = b c b R; b R, = b R b k b c bc R b = b0 = b c br b k = bc R b b k =)=) b k = b c b R 0
l br; b R T l = l h bumn m n ; bub bi l =) l bc b I l = l bc l = h 3 b c b c l l i = b c =) =) b c = l h bumn m n ; bub bi l b c = " 3 h b b U + (U ; U ) U 3 i 3 b k = p "3 h b b U U + (U ; U ) U 3 i 3 = p = ( ; + ; ; ) = p = ( ; + ; ; )
IV. b R, b R, = b0 : integrlhtosgi feltetel Teljesulniuk kell Riemn-Christoel gorbuleti tenzoroknk: (S) : b R sm = 0; S : b R sm = 0 (S) : R = ; ; + = b b b b {z } ; b p b p = 0 {z } S Guss Theorem Egregium Minrdi-Codzzi egyenletek : R = ; ; + = b b b b {z } ; b pp b pp = 0 {z } Bonnet-tetel (867): H es b kielegti Guss-Minrdi-Codzzi egyenleteket, tovbb det i0; kkor ez nnk szukseges es elegseges feltetele, hogy - merev terbeli mozgsoktol es tukrozesekt}ol eltekintve - egy es cskis egy olyn felulet letezzen, melynek es b z lpformuli br, = b R b k intgerls: b R = b R0 +
V. R; U =)=) r; : dierencilegyenlet z r helyvektorr r; = = F = R U VI. r; r; = b0 : integrlhtosgi feltetel r; r; = ; ; = m m m m = 0 r; =R U integrls: r = r 0 + 3