Propagation of Torsional Surface Waves in Non-Homogeneous Viscoelastic Aeolotropic Tube Subjected to Magnetic Field

Transcription

1 Intrnational Journal of Matrial Sin Innovations (IJMSI) 1 (1): 4-55, 13 ISSN xxxx-xxxx Aadmi Rsarh Onlin Publishr Rsarh Artil Propagation of Torsional Surfa Wavs in Non-Homognous Visolasti Aolotropi Tub Subjtd to Magnti Fild Rajnsh Kakar a, *, ShikhaKakar b a Prinipal, DIPS Polythni Collg, Hoshiarpur, Punjab, 1461, India b Faulty of Eltrial Enginring, SBBSIET Padhiana, Jalandhar, 1441, India * Corrsponding author. Tl.: ; fax: addrss:rkakar_163@rdiffmail.om ARTICLE INFO Artil history Rvisd:8 Marh Aptd:15Marh Kywords: Aolotropi Matrial, Magnti Fild, Visolasti Solids, Non-Homognous, Bssl Funtions A b s t r a t Th fft of magnti fild on torsional wavs propagating in nonhomognous visolasti ylindrially aolotropi matrial is disussd. Th dnsity and lasti onstant of th visolasti spimn ar nonhomognous. Bssl funtions ar takn to solv th problm and frquny quations hav bn drivd in th form of a dtrminant. Disprsion quation in ah as has bn drivd and th graphs hav bn plottd showing th fft of variation of lasti onstants and th prsn of magnti fild. Th obtaind disprsion quations ar in agrmnt with th lassial rsult. Th numrial alulations hav bn prsntd graphially by using MATLAB. Aadmi Rsarh Onlin Publishr. All rights rsrvd. 1 Introdution A larg amount of litratur is availabl on surfa wav in th monograph of Ewing [1]. But thr is a vry fw problms of ylindrially aolotropi lasti matrial hav bn onsidrd so far baus of th inhrnt diffiulty in solving ompliatd simultanous partial diffrntial quations. Kaliski [], Narain [3] and many othrs hav invstigatd th magntolasti torsional surfa wavs. Whit [4] has invstigatd ylindrial wavs in transvrsly isotropi mdia. Th lasti ylindrial shll undr radial impuls was studid by Mivor [5]. Cinlli [6] has invstigatd dynami vibrations and strsss in lasti ylindrs and sphrs. Pan and Hyligr [7] hav givn th xat solutions for magnto-ltro-lasti Laminats in ylindrial bnding. Th wav propagation in non-homognous magntoltro-lasti plats has bn solvd by Bin t al. [8]. Kong t al. [9] solvd th problm of thrmo-magnto-dynami strsss and prturbation of magnti fild vtor in non- 9 Pag

2 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 homognous hollow ylindr. Rntly, Kakar t al. [1, 11and 1] studid various surfa wavs in lasti as wll in visolasti mdia. In this study, th torsional wavs ar invstigatd in non-homognous visolasti ylindrially aolotropi matrial subjtd to a magnti fild. Th problm is solvd analytially by using Bssl s funtions and numrially by using MATLAB. Basi quations Th problm is daling with magntolastiity. Thrfor th basi quations will b ltromagntism and lastiity. Thrfor, th Maxwll quations of ltromagnti fild in th absn of th displamnt urrnt (in systm-intrnational unit) ar [14], (1a), (1b) (1), t (1d). t,, and ar ltri fild, magnti fild indution, prmability and 7 prmittivity of th vauum. For vauum, = 4 1 and = in SI units. Also, th trm Ohm's law is J E, (a) J is th urrnt dnsity and is a matrial ondutivity. Th Lorntz for on th harg arrirs is [14]. v (b) J ( EV B) ( E B). t Th homognous form of th ltromagnti wav quation is [14] ò, t (3a) 3 Pag

3 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 ò. t (3b) 1 1 r r r r Th dynamial quations of motion in ylindrial oordinat r,, z ar (Lov [13, 18]) srr 1 sr srz 1 u ( srr s ) TR, r r z r t (4a) sr 1 s sz sr v T, r r z r t (4b) srz 1 sz szz srz w TZ. r r z r t (4) srr, sr, srz, srr. s, sz, szz ar th rsptiv strss omponnts, TR, T, TZ ar th rsptiv body fors and uvw,, ar th rsptiv displamnt omponnts. Th strss-strain rlations ar [18] s rr 11 rr 1 13 zz, s 1 rr 3 zz, s zz 31 rr 3 33 zz, (5a) (5b) (5) s rz 44 rz, (5d) s z s r 55 z, 66 r. (5) (5f) ij lasti onstants ( ij = 1, 6). Th lasti onstants of visolasti mdium ar [1] t / // ij ij ij ij t (ij = 1, 6). (6) / // ij and ij ar th first and sond ordr drivativs of ij. Th strain omponnts ar [] 31 Pag

4 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 rr 1u, r (7a) 11 v u, r r (7b) zz z rz zz 1 w, z 11 w v, r r z 1 w u, r z 1 w, z (7) (7d) (7) (7f) Th rotational omponnts ar [] r z 11 w v, r z 11 u w, r z r 1 ( rv) u. r r (8a) (8b) (8) Equations for th propagation of small lasti disturbans in a prftly onduting visolasti solid will hav th body for in trms of ltromagnti for J (using Eq. (4)) and ar srr 1 sr srz 1 u (9a) ( srr s ) J, r r z r R t s 1 r s s z s r v J (9b), r r z r t srz 1 sz szz s rz w (9) J. Z r r z r t Lt us assum th omponnts of magnti fild intnsity ar r and z onstant. Thrfor, th valu of magnti fild intnsity is,, i (1) 3 Pag

5 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 is th initial magnti fild intnsity along z-axis and i is th prturbation in th magnti fild intnsity. Th rlation btwn magnti fild intnsity and magnti fild indution is (For vauum, 7 = 4 1 SI units.) (11) From Eq. (1), Eq. (), Eq. (3) and Eq. (1), w gt v t t (1) Th omponnts of Eq. (1) an b writtn as r 1 r, t 1, t 1 z t. (13a) (13b) (13) 3 Formulation of th problm Lt us onsidr a smi-infinit hollow ylindrial tub of radii and. Lt th lasti proprtis of th shll ar symmtrial about z-axis, and th tub is plad in an axial magnti fild surroundd by vauum. Sin, w ar invstigating th torsional wavs in an aolotropi ylindrial tub thrfor th displamnt vtor has only v omponnt. Hn, u, w v vrz (, ). (14a) (14b) (14) Thrfor, from Eq. (14) and Eq. (7), w gt, rr zz zr, (15a) z 1 v, z (15b) r 1 v v. r r (15) From Eq. (14) and Eq. (8), w gt, 33 Pag

6 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 1v r, z, (16a) (16b) z v v. r r (16) Using Eq. (14), Eq. (15) and Eq. (6), th Eq. (5) boms s s s s rr zz rz, (17a) / // 1 v v sr ( ) ( ), t t r r / // 1 v s z ( )( ). t t r (17b) (17) / // ij and ij ar th first and sond ordr drivativs of ij. For prftly onduting mdium, (i.. ), it an b sn that Eq. () boms v,, t (18) Eq. (1) and Eq. (18), th Eq. (13) boms, i v,, z (19) From th abov disussion, th ltri and magnti omponnts in th problm ar rlatd as v v,,,, t z () Using Eq. (19) and Eq. (1) to gt th omponnts of body for in trms of Gaussian systm of units as: v,, 4 z (1) Eq. (17) and Eq. () satisfy th Eq. (4a) and Eq. (4), thrfor, th rmaining Eq. (4b) boms 34 Pag

7 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 / // 1 v v / // 1 v ( ) ( ) ( )( ) r t t r r z t t r v / // 1 v v H v t ( ) ( ) r t t r r 4 z () Lt l / / l // // l m Cij ijr, Cij ijr, Cij ij r and r (3) ar onstants, r is th radius vtor and lm, ar non- / ij, ij, homognitis. // ij and From Eq. (3), w gt Eq. (17) as / // l 1 v v sr ( ) r ( ), t t r r / // l 1 v v sr ( ) r ( ), t t r r (4a) (4b) Using Eq. (3), th Eq. () boms / // l 1 v v / // l 1 v ( ) r ( ) ( ) r ( ) r t t r r z t t r m v r / // l 1 v v H v t ( ) r ( ) r t t r r 4 z (5) 4 Solution of th problm Lt v r ( ) () i z t [16] b th solution of Eq. (5). Hn, Eq. (5) rdus to ( l1) ( l1) 1 l r r r r r ( i ), / // / // 66 66i 66 H. ( ) / // 66 66i 66 (6) (7a) (7b) Eq. (6) is in omplx form, thrfor w gnraliz its solution for l and l 4.1 Solution for l For, l th Eq. (6) boms, 1 1 ( ) r r r r (8) 35 Pag

8 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 (9) 1 Th solution of Eq. (8) is v PJ Gr QX Gr ( ) { 1( ) 1( )} i z t (3) From Eq. (4) and Eq. (3) / // P ( Q sr { 66 66i 66 } { GJ( Gr) J1( Gr) { GX ( Gr) X1( Gr) r r i zt (31) 5 Boundary onditions and frquny quation Th boundary onditions that must b satisfid ar B1. Forr, (is th intrnal radius of th tub) sr r ( r ) B. For r, ( is th xtrnal radius of th tub) sr r ( r ) Whr and ar th Maxwll strsss in th body and in th vauum, rsptivly. r ( r ) Thr will b no impat of ths Maxwll strsss. Hn, r (3) ( r) On simplifiation, Eq. (18) and Eq. (3) givs ( ) i { PJ1( Gr) QX1( Gr)} i z t i( z t) Lt, Hn, Eq. (3) boms 1 r r r (33) (34) (35) Th solution of th Eq. (34) boms RJ ( r) SX ( r) (36) Whr J and X ar Bssl funtions of ordr zro. R and S ar onstants. From Eq. (37) and Eq. (4) { RJ ( r) SX ( r)} i z t ( ) (37) Th boundary onditions B1 and Bwith th hlp of th Eq. (31) and (3) turn into: 36 Pag

9 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 P{ GJ ( G) J ( G)} Q{ GX ( G) X ( G)} (38) 1 1 P{ GJ ( G) J ( G)} Q{ GX ( Ga) X ( G)} (39) 1 1 Eliminating P and Q from Eq. (38) and Eq. (39) GJ ( G) J ( G) GX ( G) X ( G) 1 1 GJ ( G) J ( G) GX ( Ga) X ( G) 1 1 (4) On solving Eq. (4), w gt th obtaind frquny quation GJ( G) J1( G) GX( G) X1( G) GJ ( G) J ( G) GX ( Ga) X ( G) 1 1 (41) On th thory of Bssl funtions, if tub undr onsidration is vry thin i.. and 3 nglting,..., th frquny quation an b writtn as (Watson [16]) 3 1 (4) H ( ) i / // 55 55i 55 / // (43) Putting th valu of in Eq. (4), th frquny of th wav an b found. Clarly, frquny is dpndnt on magnti fild. Put, (44) Th phas vloity 1 / an b writtn as H 1 4 / // 66 66i 66 (45), k i / // / // 66 66i 66, i / // (46) Th trm i.. magnti fild is ngativ in Eq. (45) whih rdus th phas vloity of torsional wav. Cas 1 Sin th pip undr onsidration is mad of an aolotropi matrial, thn 37 Pag

10 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 / // ij ij (47) Hn, from Eq. (4), Eq. (44) and Eq. (47) th frquny quation boms (48) 3 Using Eq. (45) and Eq. (46), th phas vloity is (49) H [ ] 55 H [ ] 1 (5) 66 Th trm i.. magnti fild is ngativ in Eq. (49) whih rdus th phas vloity of torsional wav. This is in omplt agrmnt with th orrsponding lassial rsults [15] Cas If th pip undr onsidration is mad of an isotropi matrial, thn, / // ij ij (51) Using Eq. (49) and Eq. (5), th phas vloity is H 1 (5) This is in omplt agrmnt with th orrsponding lassial rsults [3] 5.1Solution for l= For, l th Eq. (6) boms, 3 (3 ) ( 1 r r r r (53) 1 Putting ( r) in Eq. (53), on gt r 1 1 r r r r (54) 38 Pag

11 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 3 (55) Solution of Eq. (54) will b (Watson [16]) RJ ( r) SX ( r) (56) 1 Putting th valu of and in Eq. (55), w gt 1 { ( 1 ) ( 1 )} i z t RJ r SX r r ( ) (57) From th Eq. (4) and Eq. (56) R { 1rJ 1( 1r) ( ) J( 1r)} / // i( zt) sr ( 66 66i 66 ) S { 1rX 1( 1r) ( ) X( 1r)} (58) With th hlp of Eq. (3), Eq. (56) and boundary onditions B1 and B, w gt R S { 1J 1( 1) ( ) J( 1)} { 1X 1( 1) ( ) X( 1)} R S { 1J 1( 1) ( ) J( 1)} { 1X 1( 1) ( ) X( 1)} (59) Eliminating R and S from Eq. (58) and Eq. (59) { 1 J 1( 1 ) ( ) J( 1 )} { 1 X 1( 1 ) ( ) X( 1 )} { J ( ) ( ) J ( )} { X ( ) ( ) X ( )} (6) On solving Eq. (6), w gt { 1J 1( 1) ( ) J( 1)} { 1J 1( 1) ( ) J( 1)} { X ( ) ( ) X ( )} { X ( ) ( ) X ( )} If 1 is th root of th abov quation, thn (61) (6) { 1J 1( 1) ( ) J( 1)} { 1FJ 1 1( 1F1) ( ) J( 1F1)} { X ( ) ( ) X ( )} { FX ( F) ( ) X ( F)} F1 On th thory of Bssl funtions, if tub undr onsidration is vry thin i.. and 3 nglting,..., th frquny quation an b writtn as (Watson [18]) 1 ( ) 1 ( ) 1 1 (63) 39 Pag

12 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 H 3 3, ( ) / // 66 66i 66 ( i ). / // / // 66 66i 66 (64a) (64b) From th Eq. (6), Eq. (63) and Eq. (64), th phas vloity an b writtn as (sam as abov Eq. (45) and Eq. (46)) / // 55 55i 55 / // 66 66i 66 (65) Cas 1 Sin th pip undr onsidration is mad of an aolotropi matrial, thn / // ij ij (66) Th frquny quation is givn by { 3J 1( 3) ( ) J( 3)} { 3J 1( 3) ( ) J( 3)} { ( ) ( ) ( )} { ( ) ( ) ( )} X X 1 3 3X X 1 3 (67) (68) 1 H 55 3, 3, 3 at (69) Using Eq. (65), Eq. (66), Eq. (67) and Eq. (69), w gt (alulations ar don in th similar mannr as for th Eq. (48) to Eq. (5) for l as) (7) / 1 66 Cas If th pip undr onsidration is mad of an isotropi matrial, thn, / // ij ij (71) Th frquny quation (alulations ar don as for th l= as) is 4 Pag

13 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 { 4J 1( 4) ( ) J ( 4)} { 4J 1( 4) ( ) J ( 4)} { X ( ) ( ) X ( )} { X ( ) ( ) X ( )} H 3, 4. (7) Using Eq. (71) and Eq. (7), th phas vloity for this as is (sam as abov Eq. (45) and Eq. (46) 4 1 (73) 6 Numrial analysis Th fft of non-homognity on torsional wavs in an aolotropi matrial mad of visolasti solids has bn studid. Th numrial omputation of phas vloity has bn mad for homognous and non-homognous pip. Th graphs ar plottd for th two ass (l= and l=). Diffrnt valus of (diamtr/wavlngth) for homognous in th prsn of magnti fild and non homognous as in th absn of magnti fild ar alulatd from Eq. (49) and Eq. (65) with th hlp of MATLAB. Th variations lasti onstants and prsn of magnti fild in two urvs hav bn obtaind by hoosing th following paramtrs for homognous and non-homognous aolotropi pip. Th disprsion quations for both ass ar solvd numrially with th hlp of paramtrs. Homognous Pip Inhomognous Pip l Tabl 1.Matrial proprtis Pag

14 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 Th urvs obtaind in Figur 1 larly show that th phas vloity for homognous as wll as non-homognous as drass insid th aolotropi tub. Th prsn of magnti fild also rdus th spd of torsional wavs in visolasti solids. Ths urvs justify th rsults obtaind in Eq. (5) and Eq. (5) mathmatially givn by Narain [3] and Chandraskharaiahi [15]. W s that for homognous as whn magnti fild is prsnt and for non-homognous as whn magnti fild is not prsnt th variation i.. shap of th urvs is sam. For non-homognous as, th lasti onstants and th dnsity of th tub ar varying as th squar of th radius vtor. Homognous Pip Inhomognous Pip Tabl.Matrial proprtis l H (Gauss) 55 / Phas Vloity l= Diamtr/Wavlngth 7 Conlusion Fig.1 Torsional wav disprsion urvs Th abov problm dals with th intration of lasti and ltromagnti filds in a visolasti mdia. This study is usful for dttions of mhanial xplosions insid th arth. In this study an attmpt has bn mad to invstigat th torsional wav propagation in non-homognous visolasti ylindrially aolotropi matrial prmatd by a magnti fild. It has bn obsrvd that th phas vloity drass as th magnti fild inrass. Aknowldgmnts W ar thankful to Dr. K. C. Gupta for his valuabl ommnts. l= 4 Pag

15 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, 13 Rfrns 1. Ewing WM, Jardtzky, Prss F. Elasti wavs in layrd mdia. MGraw-Hill, Nw York Kaliski S, Ptykiwiz J. Dynami quations of motion oupld with th fild of tmpraturs and rsolving funtions for lasti and inlasti bodis in a magnti fild.prodings Vibration Problms1959; 1(): Narain S.Magnto-lasti torsional wavs in a bar undr initial strss. Prodings Indian Aadmi Sin1978; 87 (5): Whit JE, Tongtaow C. Cylindrial wavs in transvrsly isotropi mdia.journal of Aousti Soity1981;7(4): Mivor IK. Th lasti ylindrial shll undr radial impuls. ASME J. Appl. Mh. 1966;33: Cinlli G. Dynami vibrations and strsss in lasti ylindrs and sphrs.asme J. Appl. Mh. 1966;33: Pan E, Hyligr PR. Exat solutions for Magnto-ltro-lasti Laminats in ylindrial bnding.int. J. of Solid and Strut. 5; 4: Bin W, Jiangong Y, Cunfu H. Wav propagation in non-homognous magntoltro-lasti plats.j. of Sound and Vib. 5; 317: Kong T, Li DX, Wang X. Thrmo-magnto-dynami strsss and prturbation of magnti fild vtor in non-homognous hollow ylindr.appl. Mathmatial Modling9; 33: Kakar R, Kakar S. Propagation of Rayligh wavs in non-homognous orthotropi lasti mdia undr th influn of gravity, omprssion, rotation and magnti fild.journal of Chmial, Biologial and Physial Sins 1; (1): Kakar R, Kakar S.Influn of gravity and tmpratur on Rayligh wavs in nonhomognous, gnral visolasti mdia of highr ordr.intrnational Journal of Physial and Mathmatial Sins 13;4(1): Kakar R, Kakar S. Rayligh wavs in a non-homognous, thrmo, magnto, prstrssd granular matrial with variabl dnsity undr th fft of gravity.amrian Journal of Modrn Physis 13; (1): Lov AEH. Som Problms of Godynamis. Cambridg Univrsity prss Thidé B.Eltromagnti Fild Thory. Dovr Publiations Pag

16 Rajnsh Kakart al. / Intrnational Journal of Matrial Sin Innovations (IJMSI) Vol.1 (1): 4-55, Chandraskharaiahi DS. On th propagationof torsional wavs in magntovisolasti solids. Tnsor, N.S. 197;3: Watson GN. A tratis on th thory of Bssl funtions. Cambridg Univrsity Prss, Sond Edition Grn AE. Thortial Elastiity. Oxford Univrsity Prss Lov AEH. Mathmatial Thory of Elastiity. Dovr Publiations. Forth Edition Timoshnko S. Thory of Elastiity, MGraw-Hill Book Company. Sond Edition Wstrgaard HM. Thory of Elastiity and Plastiity. Dovr Publiations Christnsn RM. Thory of Visolastiity. Aadmi Prss Pag