Natural Frequencies of Magnetoelastic Longitudinal Wave Propagation. in an Orthotropic Circular Cylinder

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1 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 Ntul Feuenies of Mgnetoelsti Longitudinl ve opgtion in n Othotopi Ciul Cylinde * Ao-el-nou N. Ad-ll nd Aishh Rizh 3 Deptment of Mthemtis Fulty of Siene Jzn Univesity Jzn Sudi Ai Deptment of Mthemtis Fulty of Siene Sohg Univesity Sohg Egypt. 3 Deptment of Mthemtis Siene College King Khlid Univesity Ah Sudi Ai * E-mil ddess: oelnoudll@yhoo.om Astt In this ppe we study the longitudinl wve popgtion in pefetly onduting elsti iul ylinde in the pesene of n xil initil mgneti field. The elsti ylinde is ssumed to e mde of n othotopi mteil. The polem is epesented y the eutions of elstiity tking into ount of the effet of the mgneti field s given y Mxwell's eutions in the usi-stti ppoximtion. The stess fee onditions on the inne nd oute sufes of the hollow iul ylinde e used to fom feueny eution in tems of the wvelength the ylinde dii nd the mteil onstnts. Numeil lultions e otined nd the esults e epesented gphilly. It is oseved tht the longitudinl elsti wves in solid ody popgting unde the influene of supeimposed mgneti field n e diffeent signifintly fom tht of those popgting in the sene of mgneti field. Also elsti wves my onvey infomtion on eletomgneti popeties of the mteil: fo exmple though peise mesuement of the sufe uent indued y the pesene of the mgneti field. Finlly some of the elie esults e dedued s ptiul ses. Keywods: Ntul feuenies Mgnetoelstiity Longitudinl wve Othotopi mteils. Intodution Longitudinl wves e wves tht hve vitions long o pllel to thei dietion of tvel; tht is wves in whih the motion of the medium is in the sme dietion s the motion of the wve. The study of wve popgtion ove ontinuous medi is of ptil impotne in the field of engineeing mediine optis siene seismology oustis nd in spe siene. ith the dvnement of spe eseh it hs eome neessy to otin deep insight in the ehvio of mteils espeilly of the nisotopi ones tht e so feuently used in the missiles nd othe llied systems. ithout tking the onsidetion of the effet of the mgneti field the nlysis of longitudinl wve popgtion in nisotopi nd homogeneous iul ylindil shell oding to the theoy of elstiity hve een done y mny uthos: [ 3 4 5]. Moeove the popgtion of hmoni wves in iul ylindes whih e mde of isotopi o nisotopi mteils hve een investigted nd evluted numeilly on the sis of the theoy of elstiity y Misky [6] Tsi [7] nd hite nd Tongtow [8]. Among mny impotnt polems whih e onsideed in suh studies the polems of elsti wve popgtion in the pesene of stedy mgneti field hve investigted when the mteil ws isotopi homogeneous y Andeou et l. [9] Ds et l. [] Goukishw [] i [] Suhui [3]. Some of the nlogous esults on mgnetoelsti 7

2 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 wves popgtion polems ut in n nisotopi medium wee otined y Ad-ll [4 5] Dtt [6]. Genel detils nd mny efeenes on these sujets my e found in monogphs pulished y: Eingen et l. [7-8] Auld [9] Moon [] nd Nowki []. Reently the intetion of eletomgneti fields with the motion of defomle solid is eing eeiving gete ttention y mny investigtos. Theefoe mny esehes investigted the effet of the mgneti field on the wve popgtion in nisotopi ylindil mteils suh s: Bkti nd hupnsk [] studied the effets of pulsed eletomgneti fields on the dynmi mehnil esponse of eletilly ondutive nisotopi pltes. Dinzt nd S [3] pesented numeil investigtions into mgneto-eleto-elsti moduli esponsile fo the mgnetoeleti oupling s funtions of the volume etion nd hteistis of the oted inlusions. Akovet l. [4] studied tosionl wve dispesion in thee-lyeed sndwih hollow ylinde with finite initil stins. Chttopdhyy et l. [5] studied the popgtion of hoizontlly polized she wves in n intenl mgnetoelsti monolini sttum with iegulity in lowe intefe. Tng nd u [6] employed the method of eigenfuntion expnsion to solve the polems of tnsient tosionl vition esponses of finite semi-infinite nd infinite hollow ylindes. Ahy et l. [7] investigted the effet of the tnsvese isotopy nd mgneti field on the intefe wves in onduting medium sujet to the initil stte of stess of the fom of hydostti tension o ompession. etov et l. [8] foused on the ntue of feomgneti esonne FMR unde the influene of ousti osilltions with the sme feueny s FMR. Mol henko et l. [9] onstuted two-dimensionl nonline mgnetoelsti model of uent-ying othotopi shell of evolution tking into ount of finite othotopi ondutivity pemeility nd pemittivity. Ad-All nd Ao-Dh [3] studied the influene of the visosity on efletion nd eftion of plne she elsti wves in two mgnetized semi-infinite medi. Selim [3] showed the effet of dmping on the popgtion of tosionl wves in n initilly stessed dissiptive inompessile ylinde of infinite length. Di nd ng [3] illustted n nlytil method to solve mgneto-elsti wve popgtion nd petution of the mgneti field veto in n othotopi lminted hollow ylinde with ity thikness. Liu nd Chng [33] investigted the intetive ehvios mong tnsvese mgneti fields xil lods nd extenl foe of mgneto-elsti em with genel oundy onditions. In this study n ttempt hs een mde to investigte the longitudinl wve popgtion in n othotopi iul ylinde pemeted y mgneti field. The feueny eutions hve een deived in the fom of deteminnt involving Bessel funtions nd its oots give the vlues of the hteisti iul feueny pmetes of the fist thee modes fo vious geometies. These oots whih oespond to vious mode hve een veified numeilly nd epesented gphilly in diffeent vlues fo the mgneti field. Finlly some of the elie esults e dedued s ptiul ses.. Bsi Eutions The eutions of motion fo pefet onduting elsti solid in unifom mgneti field e []: τ f ρ& u& ji j i i ij3 follows: τ ij is the mehnil stess tenso ρ is the mss density of the mteil f i is Loentz foe nd given s µ o f [ h] H 4 π 8

3 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 h u 3 H o H o H o 4 nd H o is the intensity of the unifom xil mgneti field h smll petution of the mgneti field µ o is the mgneti pemeility in the medium. Fom 3 nd 4 h my e witten s: u u u h H o e H o ez z 5 Using 4 nd 5 in Loentz foe eomes: f u u u u z 6 e α o µ oh 4πρ Mxwell's eutions in this study my e witten s in Gussin units: v 4π B v v H J E B D ρe 7 t v v v H B E J denote espetively the mgneti field intensity mgneti indution eleti field intensity nd uent density vetos is the veloity of light in vuum nd the eleti field intensity is given s the fom H u E o o µ θ [ ] 8 t Eletomgneti eutions in vuum e: t v v v * * * * * * h E ul h E E h 9 θ The stin omponents e given in tems of the displements y: u u w e e θθ e zz z 9 v t v v

4 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 e z u w [ ] e θ e θ z. z e ij denote the stin omponents. Fo n othotopi elsti ody the Cuhy stess omponents e given in tems of independent elsti onstnts ij s follows u u w τ 3 z u u w τ zz z u u w τ θθ 3 z w u τ z z τ z θ τ θ. Sustituting 6 nd into one my get the eutions of motion in tems of the displements omponents s: u u 3. Fomultion of the olem u 3 w z u w z z u u u u u ρ z t u u w w w w ρ 3 z z z t [ 3] 33 Longitudinl wve popgtion in iul ylinde of tetgonl elsti mteil of inne nd oute dii nd sujeted to n xil mgneti field is onsideed. The ylinde is teted s pefet onduto nd the egions inside nd outside the elsti mteil e ssumed to e vuum. e ssume tht wves e hteized y the displement omponents in the dil nd xil dietions only. The displement field in this se in ylindil oodintes θ z is given y u u z t v w w z t 4 u v w e the displement omponents in the dil iumfeentil nd xil dietions espetively nd ll othe untities involved e funtions of z nd t only t denotes the time. 4. Solution of the olem 4.. Hmoni solutions: e now onside the popgtion of n infinite stin of sinusoidl wves long hollow iul ylinde of infinite extent suh tht the displement t eh point is smple hmoni funtion of z nd t. Theefoe we shll seek the solution of the eutions of motion nd follow the sme poedue s in Misky [6]: dφ u z t os λ d λt z w z t φ sin t z 5

5 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 π φ φ is the wve nume l is the wvelength λ is the ngul feueny nd is n l ity onstnt to e detemined lte in the nlysis. utting E. 5 in 3 nd 4 one otins: d φ dφ ρλ 3 φ d d E. 6 is onsistent with 7 povided tht is hosen to stisfy the eutions ρλ 33 3 ρλ ρλ 6 d φ dφ 33 φ d d Eliminting fom 8 we find tht p stisfies the eution: A p Bp C 9 A B C [ ρλ ρλ ρλ 33 If p nd p e the oots of this eution the oesponding funtions φ φ φ φ stisfy the eutions: d φ [ dφ d φ ] [ dφ p φ ] pφ d d d d B D B D ; D B 4AC A A The genel solutions of Es. e φ A B φ A B A B A nd B e onstnts of integtion nd fo evity denote the Bessel funtion J o I nd denote the Bessel funtion Y o K oding to the signs of p nd p. The displement field my now e witten s

6 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 dφ [ dφ u ]os λt z w [ φ φ ]sin λt z d d 3 ρλ ρλ Solution of Eleti field intensity in vuum * The genel solution of E θ fom 4 tke the fom E * A3 ksin λt z B3 ksin λt z 5 k λ / A 3 nd B3 e ity onstnts nd fo evity denotes the Bessel funtion Y o K oding to the signs of k Boundy onditions: Fo fee motion the oundy onditions e euied fo the totl stess to e vnished nd the ontinuity of the eleti field on the sufes i.e. τ z M τ M E E * z M M * * z on 6 τ τ z e the omponents of the mehnil stesses M z M e the omponents of Mxwell's stesses in the medium nd * * M M z e Mxwell's stesses in vuum. Eliminting A B B A3 B3 A fte pplying the oundy onditions 6 we get the deteminnt must e vnished leding to the following feueny eution dispesion eltion s: ij i j...6 7

7 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No k k k H πλ k k k H πλ πλ k H πλ k H

8 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 λh 5 λh 53 λh 5 λh k 56 λh 6 λh 63 λh 6 λh t J nd t I. 66 k. 5. Rdil nd Axil Vitions As the wve nume i.e. fo infinite wvelength the following simplifitions hve een mde y using the esult of [6]: ρλ λ ρλ k ρλ 3 3 [ ] [ ] [. nd the hteisti eution 7 my e witten s the podut of two deteminnts 9 4

9 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No The elements e given y 8 with. The eution ij epesents motion involving the dil displement u only oesponding to the dil vitions [5]. displement w only oesponding to the xil-she vitions [6] nd [3]. epesents motion involving the xil 6. The Numeil Clultions Fo numeil lultions we onside the following tnsfomtions: λ Ω λ i γ γ λ i β γ ρ π π Ω γ Ω h m. l l The lultions of the oots of the feueny eution 7 epesent mjo tsk nd euie the extensive effot fo numeil omputtion. Clultions hve een ied out fo the se of Titnium dioxide Rutile TiO whih elongs to the tetgonl system ystl symmety fo it is 4/mmm. It hs 6 elsti onstnts [9]. Also the density is 6.6 dyne / m dyne / m 7.33 dyne / m.39 dyne / m dyne / m dyne / m 3 ρ 4.6 gm / m the veloity of light is 3 m / se nd the pemeility is µ Guss Oested 3 m / se. o / 7. Disussion nd Conlusion The dimensionless feueny spetum Ω fo the longitudinl vitions s funtion of the tio thikness h / fo the vlue of non-dimensionl wve nume m is lulted nd given in fom of gphs. The vlues of the effetive pimy mgneti field Ho e hosen s H o Oested. The feueny eution is solved numeilly nd fo this pupose mtix deteminnt omputtion outine is used fo diffeent Ω nd h long with oot finding method to efine steps lose to its oots. Fo eh pi Ω nd h Es. 7 nd 3 e solved y using "intevl hlving" itetion tehniue [34]. The esults in these ses e pesented in the Figues -9 to illustte the effets of the 5

10 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 pimy mgneti field on the longitudinl vitions of n othotopi iul ylinde. It is le fom Figue. The fist mode of dimensionless feueny Ω deeses s the tio thikness h ineses fo 5 6 o H Oested. Howeve it ineses monotonilly s funtion of h fo the vlue of pimy mgneti field 7 ineses H o Oested. The sme ehvio is oseved fo the se of 6 H o Oested nd it is shown in Figues nd 3 fo the seond nd thid modes of dimensionless feueny Ω. Also in this se the effets of the 5 pimy mgneti field when 6 H o Oested e vey smll nd the uves e lmost identil. In Figue 4 ompison etween the fist thee modes of the feueny Ω vesus diffeent vlues of h fo Ho 7 is illustted. Futhemoe ou numeil lultions show tht ll the mode of the feueny Ω is not sensitive to the pimy mgneti field H o less thn eome less thn 5 Oested. So fo the vlues of H o less thn 3. 5 it n e negleted s thei eltive vitions It is lified tht when m the feueny eution 7 degenetes into two independent eutions: i One of them is fo unoupled dil vitions whih ontins the dil displement u only. ii The seond shows xil she vitions whih ontins the xil displement w only. The fist seond nd thid modes of the dimensionless feueny Ω s funtion of the h of dil vitions fo vious vlues of 6 H o 5 e pesented in Figues 5 6 nd 7 espetively. Futhemoe in the sme se ompison etween the fist thee modes of the feueny Ω s funtion of h when ll modes e inese when inesing the imposed mgneti field 6 H o 5 is shown in Figue 8. It is visile tht in this se H o. Figue 9 epesents the fist thee modes of dimensionless feueny Ω of xil she vitions ginst the vition of h when m. It ws found tht in this seond speil se the feueny Ω of xil she vitions is not ffeted with the vlues of the pimy mgneti field H o. Finlly some existing esults in the litetue e onsideed s the speil se of this study fo exmple Refs. [ ]. Refeenes [] Sinil N. nd Reg G. 8 "Nonline longitudinl/tnsvesl modl intetions in highly extensile suspended les" J. Sound nd Vition 3 pp.3-4. [] Honv F. Enjilel E. Sinli A.N. nd Minezmi S. A. 7 "ve popgtion in tnsvesely isotopi ylindes" Int. J. solid nd Stutues pp [3] Andeson S.. 6 "Highe-ode od ppoximtions fo the popgtion of longitudinl stess wves in elsti s J. Sound nd Vition" 9 pp [4] Ting T.C.T. 6 "Longitudinl nd tnsvese wves in nisotopi elsti mteils" At Mehni 85 pp47-64.

11 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 [5] Akov S.D. nd Guz A.N. 4 "Axisymmeti longitudinl wve popgtion in pe-stessed ompound iul ylindes" Int. J. Eng. Si. 4 pp [6] Misky I. 965 "ve popgtion in tnsvesely isotopi iul ylindes" pt I: Theoy t II: Numeil esults J. Aoust. So. Am. 37 pp.6-6. [7] Tsi Y.M. 99 "Longitudinl motion of thik tnsvesely isotopi hollow ylinde" Jounl of essue Vessel Tehnology 3 pp [8] hite J.E. nd Tongtow C. 98 "Cylindil wves in tnsvesely isotopi medi" J. Aousti So. Am. 74 pp [9] Andeou E.nd Dssios G. 997 "Dissiption of enegy fo mgnetoelsti wves in ondutive medium" Qut. Appl. Mth. 55 pp [] Ds N.C. nd Bhtthy S.K. 978 "Axisymmeti vitions of othotopi shells in mgneti field" Ind. J. ue Appl Mth. A45 pp [] Goukishw O. 99 " opgtion of wves in mgnetoelsti medi" J. Appl. hys. 67 pp [] i G. 967 "Mgneto-elstiity nd mgneto-themo-elstiity" In Advnes in Applied Mehnis Vol Ademi ess New Yok pp.73-. [3] Suhui E.S. 965 "Smll tosionl osilltions of iul ylinde with finite eletil ondutivity in onstnt xil mgneti field" Int. J. Engng. Si.. [4] Ad-ll A.N. 994 "Tosionl wve popgtion in n othotopi mgnetoelsti hollow iul ylinde" Appl. Mth. Comput. 63 pp [5] Ad-ll A.N. 997 "Mgneto-elsti dil vitions of tnsvesely isotopi hollow ylinde" Jpn Jounl of Industil nd Applied Mthemtis JJIAM 43 pp [6] Dtt B.K. 985 "On the stesses in the polem of mgneto-elsti inttion on n infinite othotopi medium with ylindil hole" Ind. J. Theo. hy. 334 pp [7] Eingen A.C. nd Suhui E.S. 975 "Elstodynmis" Volume II Ademi ess New Yok. [8] Eingen A.C. nd Mugin G.A. 99 "Eletodynmis of ontinu" Vols. Spinge New Yok. [9] Auld B.A. seond ed. 99 Aousti Fields nd ves in Solids vols. nd Keige Ml FL. [] Moon F.C. 985 "Mgneto-solid mehnis" John iley & Sons New-Yok. [] Nowki. 979 "Mgnetoelstiity" Chpte II in Eletomgneti Intetions in Elsti Solids edited y kus Spinge Vienn pp [] Bkti A. nd hupnsk O.I. "Anlysis of the effets of pulsed eletomgneti field on the dynmi esponse of eletilly ondutive omposites" Appl. Mth. Modelling doi:.6/j.pm...33 [3] Dinzt F. nd S H. "Mgneto-eleto-elsti oted inlusion polem nd its pplition to mgneti-piezoeleti omposite mteils" Intentionl Jounl of Solids nd Stutues 48 pp [4] Akov S. D. Kepele T. nd Met Egilmez M. "Tosionl wve dispesion in finitely pestined hollow sndwih iul ylinde" Jounl of Sound nd Vition pp Chttopdhyy A. Gupt S. nd Shu S. A. "Dispesion eution of mgnetoelsti she wves in [5] iegul monolini lye" Applied mthemtis nd mehnis 35 pp [6] Tng L. G. nd u. M. " Tnsient tosionl vition esponses of finite semi-infinite nd infinite hollow ylindes" Jounl of Sound nd Vition 398 pp

12 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 [7] Ahy D.. Roy I. nd Sengupt S. 9 "Effet of mgneti field nd initil stess on the popgtion of intefe wves in tnsvesely isotopi pefetly onduting medi" At Meh pp [8] etov V.M. itsev V.V. nd Sinivsn G. 9 " Mgnetoousti esonne in feite-feoeleti nnopills" Eu. hys. J. B 7 pp [9] Mol henko L.V. Loos I. I. nd Indiminov R. Sh. 8 " Detemining the stess stte of flexile othotopi shells of evolution in mgneti field" Intentionl Applied Mehnis 8 pp [3] Ao-el-Nou N. Ad-All nd S.M. Ao-Dh 8 "The influene of the visosity nd the mgneti field on efletion nd tnsmission of wves t intefe etween mgneto-visoelsti mteils" Meni 43 pp [3] Selim M. M 7 "Tosionl wves popgtion in n initilly stessed dissiptive ylinde" Applied Mthemtil Sienes 9 pp [3] Di H.L. ng. 6 "Mgnetoelstodynmi stess nd petution of mgneti field veto in n othotopi lminted hollow ylinde" Intentionl Jounl of Engineeing Siene pp [33] Liu M.F. nd Chng T.. 5 "Vition nlysis of mgneto-elsti em with genel oundy onditions sujeted to xil lod nd extenl foe" Jounl of Sound nd Vition 88- pp [34] Ro S.S. "Applied numeil methods fo enginees nd sientists" eson In. U.K. Figue. The fist mode of dimensionless feueny Ω fo longitudinl vitions vesus diffeent vlues of h/ fo diffeent vlues of H o when m. 8

13 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 Figue. The seond mode of dimensionless feueny Ω fo longitudinl vitions vesus diffeent vlues of h/ fo diffeent vlues of H o when m. Figue 3. The seond mode of dimensionless feueny Ω fo longitudinl vitions vesus diffeent vlues of h/ fo diffeent vlues of H o when m. 9

14 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 Figue 4. The fist thee modes of dimensionless feueny Ω of longitudinl vitions vesus diffeent vlues of h/ fo H o 7 when m. Figue 5. The fist mode of dimensionless feueny Ω of dil vitions vesus diffeent vlues of h/ fo diffeent vlues of H o when m.

15 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 Figue 6. The seond mode of dimensionless feueny vesus diffeent vlues of h/ fo diffeent vlues of Ω of dil vitions H o when m. Figue 7. The thid mode of dimensionless feueny Ω of dil vitions vesus diffeent vlues of h/ fo diffeent vlues of H o when m.

16 Mthemtil Theoy nd Modeling ISSN pe ISSN 5-5 Online Vol. No.9 Figue 8. The fist thee modes of dimensionless feueny Ω of dil vitions vesus diffeent vlues of h/ fo 6 H o 5 when m. Figue 9. The fist thee modes of dimensionless feueny Ω of xil she vitions vesus diffeent vlues of h/ when m.

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