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1 Available online at Intenational ejounals ISSN Intenational ejounal of Mathematics and Engineeing 49 (00) RADIAL VIBRATIONS IN MICRO ELASTIC HOLLOW SPHERE T. See Lakshmi and K. Sambaiah, Depatment of Mathematics, Kakatiya Univesity, Waangal , A.P., India seelakshmithumma@gmail.com Abstact : The fequency equations ae deived fo the adial vibations in micopola elastic hollow sphee. It is inteesting to obseve that a new type of wave is popagated which is not found in classical theoy of elasticity. The fequency equation of the classical case is obtained as a paticula case of this pape. [Key Wods: Radial Vibations Mico elastic Sphee] Intoduction The mechanical behaviou of elastic mateials with mico-stuctue has been the subject of intensive studies in ecent yeas. The theoy of micopola elasticity is fomulated by Eigen [] and stems fom the non-linea theoy of mico-elastic solids which was fomulated by Eigen and Suhubi []. In the micopola theoy, a volume element v is assumed to be a collection of micoelements Δv α α =,,...N. In addition to the classical defomation it consides the otation of mico elements about the cente of mass of v. The poblems of adial vibations of isotopic elastic sphee and hollow sphee ae discussed by Ghosh [3], Love s [4] teatise contains an account of the foced vibations of a sphee due to body foces deivable fom a potental. Love [5] consideed the sphee poblem in connection with the poblems of geodynamics. Gey and Eingen [6] obtained the complete solution of sphee subject to dynamic suface tactions and computed the natual fequencies of the fee oscillations. In this pape, we discussed the adial vibation in micopola elastic hollow sphee and obtained the fequency equations. It is inteesting to obseve that an additional fequency equation is obtained which is not encounted in the classical elasticity. Futhe, the esult of classical case is obtained as a paticula case of it.

2 T. See Lakshmi and K. Sambaiah, Intenational ejounal of Mathematics and Engineeing 49 (00) Basic equations the following. The fundamental equations fo the motion of a micopola elastic solid ae given by i) The balance of momentum equation: ii) + u + + k u + k + f - u = 0 () l,l k k,ll klm m,l k k The balance of the stess moment equation: + + k u k l - j = 0 () l,l k k,ll klm m,l k k k in the above equations, u is the displacement vecto, is the mico otation vecto, f is the body foce, l is the body couple vecto, is the density, j is the mico-inetia, an index (say k) following a comma indicates diffeentiation with espect to the coodinate x k, dot supeposed on a symbol denotes diffeentiation with espect to time t and,, k,,, ae the mateial coefficients which satisfy the following in equalities., k 0, k 0, k , -, 0 The stess tenso t kl and couple stess tenso m kl ae given by,,, t u u u k u (4) kl, kl k l l k l k kl m k l, k l k,l l, k (5) whee k l is the Konecke delta and k lm is the pemutation symbol. Fomulation and solution of the poblem: Conside a hollow sphee having a as adius of inne sphee (hollow sphee) and b as adius of oute sphee. We ae inteested only in adial vibations (i.e., adial displacement and adial mico-otation) and theefoe, we shall take the displacement and mico-otation vectos as u = u,te ˆ (6) =,te ˆ (7) Unde the absence of body foces and body couples, the equations of motion () in this case would educe to u u u + - u k t (3) (8)

3 T. See Lakshmi and K. Sambaiah, Intenational ejounal of Mathematics and Engineeing 49 (00) We suppose u = A' Cos pt+ u (9) whee u is function of only, is the phase and p is the angula fequency. Substituting (9) in (8) we get whee u u + - u h u 0 (0) p h () k The solution of (0) is given by Asin q Bcosq u q q () whee q = h (3) and A, B ae abitay constants Thus u,t A sin q Bcosq cos pt q q (4) The adial components of foce stess and couples stess can be obtained fom (4) and (5) and ae given by u t = + +k u (5) m = + + (6) Substituting (4) in (5) we get t = -q sin q q cosq q cosq sin q A k q cosq q sin q qsin q cosq B k Suppose 4 k s = (8) k so that (7)

4 T. See Lakshmi and K. Sambaiah, Intenational ejounal of Mathematics and Engineeing 49 (00) s = k Now the equation (7) educes to t = -q sin q q cosq s q cosq sin q A q cosq q sin q s q sin q cos q B 0 Fo fee vibations adial stess on the bounday must be zeo. Theefoe, we have t 0 at a and b (9) In view of (9), we have s h a tanh a sha A sha tanh a s h a B 0 s h b tanh b shb A shb tanh b s h b B 0 Eliminating A and B fom equations (0) and (), we get h a s has tan ha h a s tan ha sha h b s tanh b shb h b s hbs tanh b which is the fequency equation fo adial vibations coesponding to maco displacement Allowing k 0 the classical esult [3] can be obtained fom (). Unde the absence of body foces and body couples, the equations of motion (), in the pesent case educe to we suppose k j t (0) () () (3) B'cos pt (4) whee is a function of only, is the phase and p is angula fequency. Substituting (4) in (3) we get whee h 0 (5) h p j The solution of (5) is given by

5 T. See Lakshmi and K. Sambaiah, Intenational ejounal of Mathematics and Engineeing 49 (00) Csin q Dcosq q q (6) whee q h and C, D ae constants. (7) Thus Csin q Dcosq.cos pt q q (8) Substituting (8) in (6), we get q sin q q cosq q cosq sin q C (9) q cosq q cosq q sin q cosq D 0 Suppose s (30) so that s In view of (30), the equation (9) educes to q sin q qmq sq cosq sin q C q cosq q sin q s q sin q cosq D 0 Fo fee vibation adial couple stess on the bounday must be zeo. Theefoe, we have m 0 at a and b (3) In view of (3), we have s h a tanh a sha C sha tanh a s h a D 0 (3) s h b tanh b shb C shb tanh b s h b D 0 (33) Eliminating C and D fom (3) and (33), we get h a s tanh a sha h b s tanh b shb h a s h as tanh a h b s h bs tanh b (34)

6 T. See Lakshmi and K. Sambaiah, Intenational ejounal of Mathematics and Engineeing 49 (00) which is the fequency equation of adial vibation. It is inteesting to obseve that the fequency equation (34) is additional and it is due to the effect of mico otation. Futhe it is not encounteed in classical case. Refeences [] Eingen, A.C.: J.Math & Mech s , (966). [] Eingen A.C., and Suhubi E.S.: Int. J. Engg. Sci., (964). [3] Ghosh.P.K.: The Mathematics of waves and vibation. The MacMillon Company of India Limited, India, (975). [4] Love A.E.H.: A teatisc on the Mathematical theoy of elasticity 4 th edition. Dove, New Yok (944). [5] Love A.E.H.: Some poblems of Geodynamics, Cambidge Uni. Pess, London and New Yok (96). [6] Gey R.M. and Eingen A.C.: The elastic sphee unde dynamic and impact loads, ONR Tech. Rep. No.8, Pudue Univ. Lafayette, Indiana (955)