International Journal of Pure and Applied Sciences and Technology

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1 Int. J. Pue l. Sci. Technol. () (0). -6 Intentionl Jounl of Pue nd lied Sciences nd Technology ISSN vilble online t Resech Pe Rdil Vibtions in Mico-Isotoic Mico-Elstic Hollow Shee R. Sinivs * M.N. Rjshek nd. Smbih Detment of Mthemtics BITS Nsmet Wngl- 06.P. Indi Detment of Mthemtics JNTUH College of Engineeing imng- 00.P. Indi Detment of Mthemtics ktiy Univesity Wngl P. Indi * Coesonding utho e-mil: (emidi_sinivs@yhoo.co.in) (Received: 8--; cceted: --) bstct: The fequency equtions e deived fo the dil vibtions in mico-isotoic mico-elstic hollow shee. It is inteesting to obseve tht two dditionl fequencies e found which e not encounteed in clssicl theoy of elsticity. The esult of the clssicl cse is obtined s ticul cse of it. eywods: Rdil vibtions Mico-Elstic hollow shee Mico-Elstic sheicl shell.. Intoduction: Eingen [] develoed the theoy of mico-mohic mteils to conside the mico stuctue of mteils s the clssicl theoy of elsticity is indequte fo the mteil ossessing gnul stuctue. In this theoy volume element V is ssumed to be mde u of sub volumes V (α) (α N) nd it consides the defomtion of V (α) bout the cente of mss of V in ddition to the defomtion of V. Polymes comosites soils ocks concete etc. e tyicl medi with micostuctue. Guthei [] found luminum-efxy comosite to be mteil ossessing mico-stuctue. The theoy of mico-mohic mteils is simlified by oh [] using the ostultes of coincidence incile diections ssuming mico-isotoy nd clled it s theoy of mico-isotoic mico-elstic mteils. Love s [6] tetise contins n ccount of the foced vibtions of shee due to body foces deivble fom otentil. Love [7] consideed the shee oblem in connection with the oblems of geodynmics. The oblems of dil vibtions of isotoic elstic shee nd hollow shee e discussed by Ghosh [] Gey nd Eingen [] obtined the comlete solution of shee subject to dynmic sufce tctions nd comuted the ntul fequencies of the oscilltions.

2 Int. J. Pue l. Sci. Technol. () (0) -6 In this e we investigted the dil vibtions in mico-isotoic mico-elstic hollow shee nd obtined the fequency equtions. It is inteesting to obseve tht two dditionl fequencies e obtined which e not encounteed in the clssicl theoy of elsticity.. This cn be ttibuted due to the effect of consideing mico stuctue of the mteil. Futhe the esults of clssicl cse e obtined s ticul cse of this e.. Bsic Equtions: The equtions of motion nd the constitute equtions of mico-isotoic mico-elstic solid without body foces nd body coules e given by Pmeshwn nd oh [8] The dislcement equtions of motion e u ( ) u m ( ) um ε kmφ k ρ t Whee B φ B φ mm kk ij ( B δ B φ B ) φ ( ij) kk m m φ ij ( δ λ σ B τ µ B τ σ φ φ ) ρj t 7 τ 0 φ ( φij ) ρj t ( ij) m () () () σ B τ τ 9 τ 7 τ 0 B τ σ nd σ B τ 9 > 0 > 0 > 0 > 0 > 0 B B > 0 B > 0 B > 0 B < B < B B > 0 B The stess coule-stess nd stess moment e s follows: () () t ( km) e km δ e (6) km t [ km] [ km] km σ ε ( φ ) (7) σ φ δ φ (8) ( km) km ( km)

3 Int. J. Pue l. Sci. Technol. () (0) -6 6 t k ( mn) Bφ kδ mn Bφ( m n) k (9) m kl ( B φ B φ B φ δ ) (0) l k k l kl Whee ρ is the vege mss density j is the mico-ineti. The mco dislcement in the mico elstic continuum is denoted by u k nd the mico defomtion by φmn fo the line theoy we hve the mco stin e km e( k m) the mco ottion vecto k ε kmnu n m the mico-stin φ ( m n) nd mico-ottion φ ε kmφkm.the stess mesues e the symmetic stess (mco-stess) t kmn the eltive stess (mico-stess) σ km nd the stess moment t kmn. lso the coule stess tenso m ε t. The symbol ( ) shows tht the quntity is symmetic nd [ ] shows the quntity is k nm kmn skew-symmetic. λ µ σ σ σ τ τ τ 7 τ 9 nd τ 0 e the ten elstic moduli.. Fomultion nd Solution of Poblem: We conside mico isotoic mico elstic hollow shee hving s dius of inne shee (hollow shee) nd b s dius of oute shee. In the esent cse we hve the mco dislcement mico-ottion nd mico-stin. We e inteested only in dil vibtions (i.e. dil dislcement dil mico-ottion nd dil mico-stin) nd theefoe we shll tke the comonents of mcodislcement mico-ottion nd mico-stin s u u(t)e () φ φ(t)e () φ φ (t) () Whee e is the unit vecto t the osition vecto in the diection of the tngent to the cuve. Unde the bsence of body foces nd body coules the equtions of motion () to () would educe to u u u ( ) t ρ u () φ φ ρj φ φ ( B B ) ( B B ) t () ρj φ B φ B φ φ φ (6) t B φ φ 0 (7) In view of (7) the eqution (6) educes to B φ φ Whee ρj φ t (8)

4 Int. J. Pue l. Sci. Technol. () (0) -6 7 The boundy conditions e u t ( ) u 0 t nd b (9) m - (B B B ) φ - B φ 0 t nd b (0) φ t () (B B ) 0 t nd b () We suose u u cos(t ) () Whee u is function of only is the hse nd is the ngul fequency. Substituting () in () we get u Whee h u u h u ρ 0 () () s the cente 0 is not oint of hollow shee the solution of () is given by u sin q B cos q q q Whee q h nd B e bity constnt In view of () the eqution () becomes u sin q B cos q cos(t ) (7) q q Substituting (7) in (9) we get () (6) nd Whee [(s-h ) tnh-sh] [sh tnh s- h ]B0 (8) [(s-h b ) tnhb-shb][shb tnhbs-h b ]B0 (9) s (0) Eliminting nd B fom equtions (8) nd (9) we get ( h h s) tnh sh s hs tnh ( h b h b s) tnh b shb s hbs tnh b ()

5 Int. J. Pue l. Sci. Technol. () (0) -6 8 which is the fequency eqution fo dil vibtions coesonding to mco-dislcement nd it is disesive. The esult of clssicl elsticity Ghosh [] cn be obtined s ticul cse of it by llowing σ nd σ tends to 0. Fo σ 0 nd σ k/ the micool esult See Lkshmi nd Smbih [9] is obtined. Now we suose φ φ cos(t ) () Whee φ is function of only is the hse nd is the ngul fequency. Substituting () in () we get φ φ φ 0 h φ () Whee h ( B ρj B B ) () s the cente 0 is not oint of hollow shee the solution of () is given by C sin q D cos q φ () q q Whee q h nd C D e bity constnts. (6) In view of () the eqution () becomes C sin q D cos q φ cos(t ) (7) q q Substituting (7) in (0) we get [( s h ) tnh sh ] C [ s h tnh s h ] D0 (8) nd b [( s h b ) tnh b sh ] C [ sh b tnh b s h b ] D0 (9) Whee s ( B ) ( B B B ) Eliminting C nd D fom (8) nd (9) we get ( h h s ) tnh s s h h s tnh ( h h b b s ) tnh s b s h b h bs tnh b (0) which is the fequency eqution fo dil vibtions coesonding to the mico ottion nd it is lso disesive. This is new wve which is not encounteed in the clssicl elsticity. The elstic constnts involved in this fequency eqution othe thn the clssicl constnts λ nd µ. The micool esult See Lkshmi nd Smbih [9] coesonds to it cn be obtined s ticul cse of it fo B α/ B β/ nd B γ/.

6 Int. J. Pue l. Sci. Technol. () (0) -6 9 Now we suose φ G()cos(t ) () whee G() is function of only is the hse nd is the ngul fequency. Substituting () in (8) we get G G l G 0 () Whee l ρj B Let G() T() () Substituting () in () we get T () T () l T() 0 Which cn be exessed s T T ( il) T 0 It is Bessel eqution whose solution is T ( ) L J ( il) LY ( il) Whee J Y e Bessel functions with imginy guments nd is witten s T ( ) LI ( l) L ( l) Whee L L e bity constnts () () Substituting () in () we get G ( ) [ LI ( l) L ( l) Hence the eqution () educes to φ [ LI ( l) L ( l)] Cos( t ε ) (6) Substituting (6) in () we get two homogeneous equtions in L nd L. Fo the existence of non zeo solutions the deteminnt of the coefficient mtix must be zeo.

7 Int. J. Pue l. Sci. Technol. () (0) Tht is 0 (7) l l Whee li I ( ) l ( ) lb lb bli I ( ) bl ( ) The exnsion of the deteminnt (7) we get bl I li li bi I bli 0 I bl I Which is the fequency eqution nd it is due to mico-stin. Futhe it is not encounteed in neithe clssicl no micool cses s the clssicl theoy consides only mco-defomtion nd the micool theoy consides mico-ottion in ddition to mco-defomtion. s the fequency eqution obtined in such comlicted fom futhe discussion in not initited.. Secil Cse: Sheicl Shell We suose the shell is bounded by shee of dius nd d whee d is smll then the equied fequency eqution is f()f(d) (9) ( h Whee f() ( h s) tnh sh s) hs tnh Using the Tylo s seies exnsion nd neglecting the second nd highe ode tems of d the eqution (9) educes to ( f ( )) 0 (8) Intoducing l h we hve ( l s ) tn l ls 0 l ( l s) ls tn l (0) Simlifying the eqution (0) we get l s s () This is the fequency eqution in thin sheicl shell. Substituting the vlues of l s nd h in the eqution () we get the fequency eqution s ρ The micool esult coesonding to () cn obtined s ticul cse of it when σ 0 nd σ k/. Futhe llowing k 0 the clssicl esult Ghosh [] cn be obtined. ()

8 Int. J. Pue l. Sci. Technol. () (0) -6 6 Similly coesonds to mico-ottion we hve nothe fequency eqution l s s () Whee l h Now substituting the vlues of l h nd s in the eqution () we get the fequency eqution s ( B )( B B ) ρj ( B B ) () Refeences [].C. Eingen Mechnics of micomohic mteils In Poceedings of the th Intentionl Congess of lied Mechnics Munich (96b) -8 Singe Belin. [] R.D. Guthei Exeimentl Investigtions on Micool Medi Wold Scientific Singoe 98. [] P.. Ghosh The Mthemtics of Wve nd Vibtion The Mc Millon Comny of Indi limited Indi 97. [] R.M. Gey nd.c. Eingen The Elstic Shee unde Dynmic nd Imct Lods ONR Technicl Reot No.8 Pedue Univesity Lfyette Indin 9. [] S.L. oh secil theoy of micoelsticity Intentionl Jounl of Engineeing Science 8(7) (970) 8-9. [6].E.H. Love Tetise on the Mthemticl Theoy of Elsticity ( th edition) Dove Publictions New Yok 9. [7].E.H. Love Some Poblems of Geodynmics Cmbidge Univesity Pess London nd New Yok 96. [8] S. Pmeshwn nd S.L. oh Wve ogtion in mico-isotoic mico-elstic solid Intentionl Jounl of Engineeing nd Science (97) [9] T. See Lkshmi nd. Smbih Rdil vibtions in micoolo elstic hollow shee Intentionl ejounl of Mthemtics nd Engineeing 9(00) 9-97.