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Avilble online t www.interntionlejournls.com ISSN 0976-4 Interntionl ejournls Interntionl ejournl of Mthemtics nd Engineering 93 (0) 85 855 RADIAL VIBRATIONS IN MICROPOLAR THIN SPHERICAL SHELL R.

Smbih*** * Dertment of Mthemtics, BITS, Nrsmet, Wrngl - 50633 ** Dertment of Mthemtics, JNTUH College of Engineering, Krimngr-50550 *** Dertment of Mthemtics, Kktiy University, Wrngl-506009 E-mil:

1 Avilble online t ISSN Interntionl ejournls Interntionl ejournl of Mthemtics nd Engineering 93 (0) RADIAL VIBRATIONS IN MICROPOLAR THIN SPHERICAL SHELL R.Srinivs*, M.N.Rjshekr** nd K.Smbih*** * Dertment of Mthemtics, BITS, Nrsmet, Wrngl ** Dertment of Mthemtics, JNTUH College of Engineering, Krimngr *** Dertment of Mthemtics, Kktiy University, Wrngl E-mil: remidi_srinivs@yhoo.co.in Abstrct : The frequency equtions re derived for the rdil vibrtions in microolr elstic thin shericl shell. It is interesting to observe tht new tye of wve is rogted which is not found in clssicl theory of elsticity. The frequency eqution of the clssicl cse is obtined s rticulr cse of this er. Key Words: Rdil vibrtions, Micro olr elstic shericl shell INTRODUCTION In the micro olr theory, volume element V is ssumed to be collection of micro elements V ( ) ( =,,.N). In ddition to the clssicl deformtion it considers the rottion of micro elements bout the centre of mss of V. The mechnicl behvior of elstic mterils with micro-structure hs been the subject of intensive studies in recent yers. The theory of micro olr elsticity is formulted by Eringen [] nd stems from the non-liner theory of micro-elstic solids which ws formulted by Eringen nd Suhubi []. The roblem of rdil vibrtions of isotroic elstic shere nd hollow shere re discussed by Ghosh [3], Love s [5] tretise contins the forced vibrtions of shere due to body forces derivble from otentil. Love [6] considered the shere roblem in connection with the roblems of geodynmics. Grey nd Eringen [4] obtined the comlete solution of shere subject to dynmic surfce trctions nd comuted the nturl frequencies of the free oscilltions. T. Sree Lkshmi nd K.Smbih [7] obtined the frequency equtions for rdil vibrtions in micro olr elstic hollow shere.

2 Interntionl ejournl of Mthemtics nd Engineering 93 (0) R.Srinivs*, M.N.Rjshekr** nd K.Smbih*** In this er, we discussed the rdil vibrtions in micro olr thin shericl shell nd obtined the frequency equtions. It is interesting to observe tht n dditionl frequency eqution is obtined which is not encountered in the clssicl elsticity. Further the result of clssicl cse is obtined s rticulr cse of it. The equtions re reduced to nondimensionl forms nd grhs re drwn by ssuming certin vlues of non-dimensionl quntities. BASIC EQUATIONS The fundmentl equtions for the motion of microolr elstic solid re given by the following. (i) (ii) The blnce of momentum eqution ( + )u l,lk + ( + k)u k,ll + k klm m,l + (f K ü k ) = 0 () The blnce of the stress moment eqution ( + ) l,lk + k,ll + k klm u m,l k k + ( l k - j.. k) = 0 () In the bove equtions, u _ is dislcement vector, f is the body force, l is the body coule vector, is the density, j is the micro inerti, n index (sy k) following comm indictes differentition with resect to the coordinte (X K ), dot suerosed on symbol denotes differentition with resect to the time t nd,, k,,, re the mteril coefficients which stisfy the following inequlities k 0, + k 0, k , -, 0 (3) The stress tensor t kl nd coule stress tensor m kl re given by t kl = u r,r kl + (u k,l +u l,k ) + k(u l,k - klr r) (4) m kl = r,r kl + k,l + l,k (5) where kl is the kronecker delt nd klm is the ermuttion symbol. FORMULATION AND SOLUTION OF THE PROBLEM The frequency equtions of rdil vibrtions in microolr elstic hollow shere re given by [7] ( h s)tnh sh = ( h b s ) tnh b shb ( h s) hs tnh ( h b s) hbs tnhb (6) nd ( h ( h s ) tnh s ) h s s h tnh = ( h ) tnh b s b sh b ( h b s ) h bs tnh, b (7) 85

3 Interntionl ejournl of Mthemtics nd Engineering 93 (0) R.Srinivs*, M.N.Rjshekr** nd K.Smbih*** where h s k 4 k k,, h, s, j is frequency, nd b re the rdii of inner nd outer shere of hollow shere. Now, we suose the shell is bounded by shere of rdius nd + d, where d is smll, then the required frequency eqution is f () = f( +d) (8) where f () = ( h ( h s) tnh hs s) hstnh Using the Tylor s series exnsion nd neglecting the second nd higher order terms of d then the eqution (8) reduces to Introducing h = l we hve ( f ( )) 0 (9) l ( l ( l s )tnl ls s) lstnl 0 (0) Simlifying the eqution (0) we get l = 3s s () This is the frequency eqution in thin shericl shell. Substituting the vlues of l, s nd h in the eqution () we get the frequency eqution s 4 k 3 k () k Allowing k 0, the clssicl result [3] cn be obtined. Similrly, corresonds to micro rottion we hve nother frequency eqution. l = 3s s where l = h Now substituting the vlues of l, h nd s in the eqution (3) we get the frequency eqution s (4) j ( ) The dditionl frequency (4) is not encountered in clssicl elsticity nd it corresonds to micro rottion. The eqution () reduced to non dimensionl form (4 m ) m3(3m m ) (5) ( m m ) (3) 853

Interntionl ejournl of Mthemtics nd Engineering 93 (0) 85 855 R.Srinivs*, M.N.Rjshekr** nd K.

4 Interntionl ejournl of Mthemtics nd Engineering 93 (0) R.Srinivs*, M.N.Rjshekr** nd K.Smbih*** where k m, m nd m 3 nd the eqution (4) reduced to ( m4 )( m4 3m5) (6) j( m m ) where m 4, m NUMERICAL CALCULATIONS Fig. Fig. Fig. 3 Fig. 4 Fig

5 Interntionl ejournl of Mthemtics nd Engineering 93 (0) R.Srinivs*, M.N.Rjshekr** nd K.Smbih*** The figures, nd 3 re the grhs drwn for the frequency (5) for vrious vlues of rdii for the following cses resectively. (i) m = 0., m = 0.5 nd m 3 = 0., 0.4, 0.7 (ii) m = 0., m 3 = 0.5 nd m = 0., 0.4, 0.7 (iii) m = 0.5, m 3 = 0. nd m = 0., 0.4, 0.7 It is observed tht the curves corresonding to (i) m = 0., m = 0.5, m 3 = 0. (ii) m = 0., m = 0., m 3 = 0.5 (iii) m = 0., m = 0.5, m 3 = 0. re rbolic she. In other cses, the curves re rbolic when rdius is greter thn. nd lmost stright line for rdius less thn.. The figures 4 nd 5 re the grhs drwn for the frequency (6) for vrious vlues of rdii for the following cses. (i) m 4 = 0. nd m 5 = 0.3, 0.6, 0.9 (ii) m 5 = 4 nd m 4 =,, 3 It is observed tht the curves re rbolic in nture nd the frequency decreses when rdius increses. REFERENCES []. Eringen, A.C.: Liner theory of microolor elsticity. Journl of Mthemtics nd Mechnics 5, (996). []. Eringen, A.C., nd Suhubi, E.S.: Non liner theory of simle micro elstic solids. Interntionl Journl of Engineering nd Sciences, (964). [3]. Ghosh, P.K.: The Mthemtics of wves nd vibrtions. The Mc Millon Comny of Indi Limited, Indi (975). [4]. Grey, R.M., nd Eringen, A.C.: The elstic shere under dynmic nd imct lods. ONR Tech.Re.No.8, Purdue University. Lfyette, Indin.(955). [5]. Love, A.E.H.: A tretise on the mthemticl theory of elsticity. 4 th Edition, Dover Publictions, New York (944). [6]. Love, A.E.H.: Some roblems of Geodynmic. Cmbridge University Press, London nd New York (96). [7]. T. Sree Lkshmi nd K.Smbih.: Rdil vibrtions in microolor elstic hollow shere. Interntionl e-journl of Mthemtics nd Engineering 49, (00). 855

International Journal of Pure and Applied Sciences and Technology

International Journal of Pure and Applied Sciences and Technology Int. J. Pue l. Sci. Technol. () (0). -6 Intentionl Jounl of Pue nd lied Sciences nd Technology ISSN 9-607 vilble online t www.ijost.in Resech Pe Rdil Vibtions in Mico-Isotoic Mico-Elstic Hollow Shee R.