MAGNETO-ELASTIC INTERACTIONS IN A CRACKED FERROMAGNETIC BODY

Transcription

1 MAGNETO-ELASTIC INTEACTIONS IN A CACKED FEOMAGNETIC BODY by Stenk rutyunyn The ubmtted to the fculty of the Vrgn Polytechnc Inttute nd Stte Unerty n prtl fulfllment of the requrement for the degree of MASTE OF SCIENCE n MATEIALS SCIENCE AND ENGINEEING Approed by: Dr. Wllm T. eynold, Chr Dr. Dreh nyn Dr. Yu Wng Noember 3, 6 Blckburg, Vrgn Keyword: Mgnetoeltcty, Ferromgnetc, Nonlner Lw of Mgnetzton, Crck

2 MAGNETO-ELASTIC INTEACTIONS IN A CACKED FEOMAGNETIC BODY Stenk rutyunyn ABSTACT The tre-trn tte of ferromgnetc plne wth mong crck h been netgted n th tudy. The model conder oft mgnetc ferroeltc body nd ncorporte reltc nonlner uceptblty. A mong crck preent n the body nd propgtng n drecton perpendculr to the mgnetc feld. Aumng tht the procee n the mong coordnte re ttonry, Fourer trnform method ued to reduce the med boundry lue problem to the oluton of pr of dul ntegrl equton yeldng to cloed form oluton. A reult of th netgton, the mgnetoeltc tre ntenty fctor obtned nd t dependency upon the crck elocty, mterl contnt nd nonlner lw of mgnetzton re hghlghted. It h been hown tht tre reult round the crck eentlly depend on eternl mgnetc feld, peed of the mong crck, nonlner lw of mgnetzton, nd other phycl prmeter. The reult preented n th work how tht when crcked ferromgnetc tructure under the nfluence of mgnetc feld t necery to tke nto ccount the ntercton effect between deformton of the body nd mgnetc feld nd tht uch ntercton cn brng to new condton for trengthenng the mterl. Cloed form oluton for the tre-trn tte re obtned, grphcl repreentton re uppled nd concluon nd propect for further deelopment re outlned.

3 ACKNOWLEDGMENTS Thnk to the NSF GFP for prodng me wth the Fellowhp for 4-7 cdemc yer to purue my grdute tudy t Vrgn Tech. Th work would not he been poble wthout the help of mny people who t my pleure to thnk here: I would lke to thnk my dor, Dr. Wllm eynold, for llowng me to work wth hm, for h long hour of luble dcuon, contnt upport nd gudnce throughout my reerch. I m grteful to my co-dor, Dr. Dreh nyn, for h contnt upport nd boe ll for h confdence n me. Wthout h help I would not he fnhed th work. I would lo lke to epre my grttude to my commttee member, Dr. Yu Wng, for greeng to be on my commttee nd lo I m thnkful for h helpful dcuon. Mny thnk to Mrko eynold, for her emotonl nd enthutc upport tht mde me feel t home n the Unted Stte. Thnk to Nrmn Aloyn for beng lwy redy to hre nd reon my concern. I m grteful to ll of my frend, for ther nluble frendhp tht went beyond boundre of lnguge nd geogrphy. Mny thnk to my ter, uznn Kzhryn, for her contnuou loe nd upport throughout my lfe. Foremot pprecton goe to my Mom, rtn Yertyn. Th work dedcted to her, for her uncondtonl loe nd ptence.

4 TABLE OF CONTENTS Abtrct Acknowledgement Tble of Content Lt of Fgure Mgneto-Eltc Intercton n Crcked Ferromgnetc Body Objecte. Introducton. Bc Equton nd Boundry Condton of Ferromgnetc Body 4. Mgnetottc nd Mwell Stre Tenor 4. The Generl Equton n n Euler Sytem of Coordnte 9.3 The Generl Equton n Lgrnge Sytem of Coordnte.4 The Lnerzed Equton nd Boundry Condton 4.5 Dfferent Mgnetc Suceptblte of Soft Ferromgnetc Mterl 8 3. Stre-Strn Stte of Ferromgnetc Plne wth Mong Crck 3. Mthemtcl Modelng 3. Goernng Equton nd Boundry Condton 3.3 Soluton Methodology Mgnetoeltc Stree nd Stre Intenty Fctor 6 4. Numercl Computton nd Dcuon 3 4. Stre Intenty Fctor 3 4. Mgnetoeltc Stree 4 Concluon 47

5 Potentl Work 48 eference 49 Append A: Nomenclture 5 Append B: The Soluton of Dul Integrl Equton 54 Append C: Mgnetzton Lw n Term of Normlzed Prmeter 56 Vt 58

6 LIST OF FIGUES Fgure : A olume element eprted by urfce of dcontnuty 7 Fgure : Geometry of the deformed body n mgnetc feld 9 Fgure 3: Dependence of mgnetc uceptblty eru mgnetc feld for pure ron 9 Fgure 4: Ferromgnetc plne wth crck n mgnetc feld Fgure 5: Dependence of normlzed tre ntenty fctor on dmenonle mgnetc feld for rou lue of prmeter wth the mgnetzton epreed n Drefou form 3 Fgure 6: Dependence of normlzed tre ntenty fctor on dmenonle mgnetc feld for rou lue of prmeter wth the mgnetzton epreed n b Lner form 34 Fgure 7: Dependence of normlzed tre ntenty fctor on dmenonle mgnetc feld for rou lue of prmeter wth the mgnetzton epreed n c ylegh form 36 Fgure 8: Dependence of normlzed tre ntenty fctor on dmenonle mgnetc feld for rou lue of prmeter wth the mgnetzton epreed n d Mgneto rgd form 38 Fgure 9: Normlzed mgnetoeltc her tre eru dmenonle mgnetc feld for rou lue of prmeter 4 Fgure : Normlzed mgnetoeltc tenle tre eru dmenonle mgnetc feld for rou lue of prmeter 44

7 MAGNETO-ELASTIC INTEACTIONS IN A CACKED FEOMAGNETIC BODY OBJECTIVE The purpoe of th reerch to netgte the effect of mgnetc feld on the tretrn tte round crck n ferromgnetc mterl. Specfclly, we wh to determne f the tre dtrbuton t crck tp cn be ltered by n ppled mgnetc feld nd to quntfy the mgnetc contrbuton to the brttle flure crter.. INTODUCTION Ferromgnetc mterl re condered good cnddte for the frt wll n mgnetc fuon energy MFE rector. Beng n preence of hrh lodng condton, ncludng therml nd mechncl lod nd of trong mgnetc feld the frt wll of MFE rector could be ubjected to tre concentrton round tructurl defect or crck. It clerly pper tht uch poblty could conttute n obtcle towrd the deelopment of th energy ource. The recent yer he wtneed ncreed nteret n netgton of the mgnetoeltcty for ferromgnetc mterl. The generl theory of mgnetoeltcty for ferromgnetc old deeloped by mny uthor [-, 4-5, 7, -3, 5, 7, ]. The theory uggeted by [, 5] cn be ppled to the netgton of the phononmgnon couplng mgnetocoutc reonnce. The couplng pronounced when the we frequency ner or boe the mgnetc reonnce frequency, whch uully

8 hgher thn 9 z for emple mgnetocoutc reonnce n mterl Yttrum-Iron- Grnett obered when the frequency of pne we z. In the pper [, 3, 8, ] theory preented for the mgnetoeltc ntercton n oft ferromgnetc mterl of mult-domn tructure when mgnetc feld ntenty ector nd mgnetzton ector M re prllel n rgd body tte: M χ χ clled the mgnetc uceptblty. Notce tht oft mgnetc mterl chrcterzed by mll hytere loe nrrow hytere loop for M cure nd low remnnt mgnetzton. Mny Nckel-Iron lloy ued wdely core mterl for motor, genertor, nductor nd trnformer re of th type. The reult relted to the mgnetoeltc tblty nd brton of thn-wlled bode, the tre deformton tte SDS nd lo the we propgton of ferromgnetc mterl wth lner lw of mgnetzton re of pecl nteret [, 6, 9-]. In lrge number of work relted to the theory of mgnetoeltcty of ferromgnetc mterl, t h been uppoed tht the mgnetc feld lnerly connected wth the mgnetzton of the mterl, e.g. oft ferromgnetc wth lner lw of mgnetzton w condered [,,3-5, 9-3]. oweer, th umpton ld for the ferromgnetc n the domn of ery wek mgnetc feld or for mot of nonferromgnetc n the trong mgnetc feld. Wth the pplcton of ferromgnetc mterl nd tructure n trong mgnetc feld,.e. hgher thn Tel uch the tructure of frt wll n fuon rector, the mgnetc feld generted n the tructure my be ner to the regon of turton of the conttute relton of mgnetc feld. In uch ce to conder non-lner dependence between the mgnetzton nd the mgnetc feld become mperte. Tht why t become nteretng to conder non-

9 lner dependence between the mgnetzton nd the mgnetc feld well ther nfluence on rou phycl procee the bucklng of thn ferromgnetc plte; the nfluence of nonlnerty of mgnetzton on tre -trn tte; we propgton nd etc.. In th work the problem of tre-trn tte of ferromgnetc plne wth mong crck dcued n detl. 3

10 . BASIC EQUATIONS AND BOUNDAY CONDITIONS OF ELASTIC FEOMAGNETIC BODY In th chpter we wll preent ll necery equton nd boundry condton for the eltc oft ferromgnetc body n n eternl mgnetc feld.. MAGNETOSTASICS AND MAXWELL S STESS TENSO We denote the mgnetc feld generted by permnent mgnetc or n nducton col n cuum by the mgnetc nducton ector B wth unt Weber/meter n MKSA ytem. When mgnetzble, eltc old plced n the feld, mgnetc moment re nduced n the body. The mgnetc moment per unt olume of deformed body clled the mgnetzton nd t denoted by M Amper/meter. Inde the body, the mgnetc nducton B whch not necerly to be equl to B. The nduced mgnetzton relted to B by B μ where Amper/meter clled the mgnetc ntenty nd M 7 μ 4π Newton/Amper unerl contnt. The ector ntroduced for conenence. In cuum, M, nd B ; n body relted to M by conttute lw. The B tfe Gu μ lw for mgnetm nd olenod, nd goerned by Ampere lw. For rgd old, the conttute lw wrtten M whch ume prtculr mple form, nd χ M for prmgnetc or dmgnetc otropc mterl. The dmenonle contnt χ clled the mgnetc uceptblty. For deformble old, depend on M well on the dplcement grdent of the body. 4

11 When body mong n mgnetc feld or when the feld chngng n tme, n electrc feld E nd current J re generted. The E relted to the B through Frdy lw. For rgd old wth electrc conducttyσ, J σ E. Vector E, B, J, nd re coupled nd complyng Mwell equton [-5, 7-8, -5, 7, 9-3]: B curle Frdy Lw t curl J Ampere Lw db Gu Lw J σ E B Generlzed Ohm Lw t where u ; u dplcement ector of mterl prtcle. t In th netgton we conder only lowly mong body n ttonry mgnetc feld. Thu, the couplng of the nduced current J nd the deformton wll not be condered here. Th certnly ld for mgnetc nultor σ. For prmgnetc conductor, th omon not jutfed, the effect of nduced current more pronounced thn tht of nduced mgnetzton. oweer, for ome of oft ferromgnetc mterl, frt ppromton, we my omt the nduced current effect. The effect of nduced current n the form of Lorentz force body force on n eltc we h been netgted by [,5]. Mny pper on tht ubject re reewed by [,3,5]. eferrng the olume V occuped by body t ny ntnt to Crten coordnte ytem, we denote the poton of mterl prtcle t tme t by X, X,. In the preence of mgnetc nducton B body re [-3, 4, 7,, 3, 5, 7]: μ X 3, the totl force F nd torque L ctng on 5

12 F μ M dv. V L μ [ M M ] dv. V oweer, the boe epreon do not mply tht wthn olume element dv, there body force per unt olume μ or body couple per unt olume μ M M becue of the ddtonl ntercton between the mgnetzton wthn the element dv nd the urroundng mgnetzed mterl. Furthermore, by pplyng rou ector formul nd mgnetottc equton, the boe ntegrl cn be trnformed nto urfce ntegrl or olume ntegrl wth entrely dfferent ntegrl. Thu on n element wth olume dv nd urfce ds, the net force eerted by eternlly ppled feld nd urroundng mgnetzed mterl cn not be unquely pecfed. The mgnetc body force f nd the body couple c ctng on the m wthn olume dv re: f μ M, c μ M.3 In the boe equton the M nd nde the mgnetzed body re relted to the eternl mgnetc nducton B through et of mgnetoeltc feld equton, conttute equton nd boundry condton gen n the uul mechncl trcton to form tre ector n t. Wth thee undertndng, mgnetoeltc tre tenor t j my be defned t n t.4 j n j where n unt ector norml to the urfce ds upon whch n t ctng. Becue of the umed etrnc body couple, the mgnetoeltc tre tenor t j not ymmetrc. 6

13 The blnce equton for lner momentum then cn be mply tted d ρ jdv, where dt tjn ds μ M dv j, S V V j j,.5 X The locl blnce equton gen n the net prgrph. The uul mechncl body force not condered here. When the olume ntegrl of the body force n.5 encloe regon whch eprted by urfce of dcontnuty Fgure, the body force dcontnuou t th urfce on ccount of the etence of urfce mgnetzton n n M M where nd - ndcte the two de of. Thu, μ r M dv μ ds M n dn μ n V Δh n ds Δh M n d M whch defned μ M n M n ds Δh.6 μ M n M n M n n ds where ue h been mde of the fct tht the tngentl component of re contnuou cro the urfce. n r Δh - Σ Fgure : A olume element eprted by urfce of dcontnuty 7

14 Another wy to elute the effect of urfce mgnetzton to ntroduce tenor T j uch tht T f where T j B j μ k kδj.7 We then cn pply the dergence theorem to chnge the olume ntegrl of f to urfce ntegrl of n Tj nd elute the jump of the norml component of T j cro the urfce, whch jut the ntegrnd n the fnl epreon of.6. The quntty T j known Mwell tre tenor. Wth the mgnetc force nd moment ntroduced n.3, there obouly n energy upply to the medum, whch equl to work done by thee force. Th h been crefully eluted n [-3, ]. The fnl reult for the rte of chnge of energy per unt olume j d M ε μ M j μρ.8 dt ρ where d dt / t j / X nd dx dt. The frt term clerly the rte of / j work done by the body force j j / f j ; the econd term the rte of work done by the body couple combned wth the rte of chnge of the mgnetzton wth repect to e tht trnlte nd rotte wth the mgnetc moment ector. 8

15 . TE GENEAL EQUATIONS IN AN EULE SYSTEM OF COODINATES Let prtcle of deformed old orgnlly t j j,,3 moed, fter deformton to X J J,,3 t tme t Fgure. The Crten Sytem of coordnte X J clled mong or Euler coordnte nd j nonmong or Lgrnge ytem of coordnte. z Z n r Y u r n r n r r r r o X B r y Fgure : Geometry of the deformed body n mgnetc feld Then the functon X, X, X 3, t.9 or ther nere decrbe the deformton for the body whole nd the prtl derte / X j or I j X / decrbe t loclly. The deformton decrbed by the 9

16 trnformton.9 to be determned from et of feld equton, boundry condton, nd conttute equton. The generl feld equton of mgnetoeltcty re dered ubttutng the body force f, body couple c defned n.3 nd the rte of energy upply ε n.8 nto the uul equton of blnce of lner momentum, ngulr momentum, nd energy, repectely. The reult, epreed n term of the current poton ector X re: dρ M conerton: ρ dt X. Lner Momentum: t j X j d j μ M ρ. X dt Angulr Momentum: t μ M. [ jk ] [ j k ] Energy chnge: du ρ dt t j j X d M μ ρ.3 dt ρ In the boe, ρ the m denty, t j the tre tenor n.4, nd U the nternl energy per unt m. Snce the energy flow due to het conducton not condered here, the rte of chrge of U the me the rte of chrge of the free energy F dcued n [7] nd [7]. The ndce wth brcket repreent the ntymmetrc prt,.e. t t t [ j] j j /. To the boe equton, we dd the mgnetc feld equton: Defnton: B μ M.4 Gu Lw: B / X.5 Ampere Lw: X.6 [ j / k ] All mgnetc quntte re to be eluted wthn body t the deformed confgurton.

17 Acro urfce of dcontnuty of deformed body, the boundry condton for B nd re the me the uul one n mgnetottc. Wth unt norml n drwn from the negte de - to the pote de of the deformed urfce of dcontnuty, the boundry condton re: n [ B ] n B B.7 n ] n.8 [ j[ k ] [ j k ] k ] The boundry condton for tree re: n [ tj Tj ] n[ tj B j μ k kδ j ] g j.9 where g j component of urfce mechncl force. The theory completed by ddng to t conttute equton for oft ferromgnetc mterl..3 TE GENEAL EQUATIONS IN LAGANGE SYSTEM OF COODINATES Let n eltc delectrc medum wth ordered mgnetc tructure be n n eternl ttonry mgnetc feld wth the mgnetc ntenty ector B nd the mgnetc nducton μ. The medum urroundng the body uppoed to be cuum. Under the nfluence of the mgnetc feld body couple per unt olume c ctng on body re: the totl force per unt olume f nd r r r r r r f μ M, c μ M.

18 The moton of the deformble ferromgnetc body under the cton of the gen mgnetc feld wll be decrbed n mong rectngulr coordnte ytem X X X 3. Intl poton of the body pont n choen coordnte ytem re defned by the Crten coordnte,, whch we lter on ued the Lgrnge nonmong coordnte 3 to the e the mong coordnte of the medum pont. Under the cton of the olume force nd the olume moment. well urfce force of nonmgnetc orgn, the medum deformed nd t moton n ytem of coordnte,, wll be decrbed by the followng equton [, 5, ] 3 t m uk uk δ mk f k ρ,. m t e t c, mk mk f k T mk,, m, k,,3 m. nd boundry condton n n e [ B B ] e [ ].3.4 t km e u [ T T ] n u δ m nk G km km m k δ m.5 m Detled derton of Eq re gen n [, 5, ]. In Eq u k re the dplcement ector component, t m the tenor of the Lgrnge mgnetoeltc tree; e mk the permutton ymbol wth e jk or ccordng to whether the ndce re n cyclc or n ntcyclc order, repectely, nd e otherwe; δ mk the Kronecker ymbol wth δ mk when m k nd δ mk jk

19 otherwe; ρ the m denty of the medum before the deformton; n k nd nk re component of the eternl norml to the undeformed nd deformed urfce of the body repectely; T km re the component of the Mwell tre tenor T mk Bm k μδ mk.6 The nde "e" nd "" here nd lter on denote cceory of the condered quntty to eternl nd nternl medum repectely. Summton tke plce by the recurrng ndce. It obou from the Eq..3 tht the tenor t m n generl ce re nonymmetrcl. It become ymmetrc only f the mgnetc moment nh c. Subttutng the lue of c k from the Eq.. nto the Eq..3 the lt of mentoned equton yeld e t μ M.7 mk m m k whence the ymmetry of the tenor t μm follow. It ey to ee tht the m m ytem of Eq not cloed nd t necery to dd to them the conttute equton of the mgnetoeltc medum, whch connect chrctertc of the deformble tte eltc deformton, tree, the mgnetc feld ntenty, nd the nducton of the mgnetc feld. The conttute equton he the form [, 7, 5, ] U U ρα,.8 t j kα je M j α j δ j u, j ε ke μ U μ, μ M / ρ.9 μ where U the nner pecfc energy per unt m; ε ke re the component of the Green deformton tenor 3

20 uk ue um um ε ke.3 e k k e ρ the m denty of the deformble medum. Ung Eq..9, the Eq..8 cn be wrtten n form U M.3 tj ρ α k α je μ ε ke j Tkng nto ccount the ymmetry of the frt term of the epreon.3 one cn ee tht the ymmetry condton.7 owng to Eq..3 re dentclly tfed. The epreon for the pecfc ntrnc energy of the deformble eltc nonconducte ferromgnetc body, ccordng to [, 7, 5, ], choen n the followng form el m U ε, M U ε U M.3 j j where el m U nd U re n eltc nd mgnetc energy, repectely. The mple rnt of ferromgnetc body the ce, when the ector nd M re prllel. e. mgnetooft ferromgnetc..4 TE LINEAIZED EQUATIONS AND BOUNDAY CONDITIONS Let u turn to the lnerzton of the mn equton nd boundry condton decrbng the behor of the deformble ferromgnetc body wth nonlner chrctertc between mgnetc feld nd mgnetzton whch re gen n the preou prgrph doptng the mn umpton of the mll deformton theory. Wth tht purpoe, we repreent the mgnetc feld chrctertc n followng form [, 7, 5, ] B B b, h, M M m.33 4

21 ere B, M nd re the mgnetc nducton ector, the mgnetzton nd the mgnetc feld ntenty n rgd tte, repectely; b m, nd h re the ddte perturbton to mentoned quntte cued by the deformton of the body. A defned boe, the quntte B, M nd re determned from the oluton of the followng mgnetottc problem: Equton n domn occuped by the body nternl domn curl r db r.34 r r r r where B μ M μ [ χ ] b Equton n domn outde of the body eternl domn e curl r e db r.35 where r B e e r e μ nd M r c Condton on the urfce of the nondeformed body r r r e r r r e n [ B B ], n [ ].36 d Condton t nfnty r r, when r /.37 B e B 3 The chrctertc of the tre-trn tte of the body the dplcement ector r component uk nd tree tenor component S m nd the quntte b m r r, nd h re beng determned from equton nd boundry condton. -.3 for the deformble ferromgnetc body. Aumng tht the deformton nd the module of 5

22 quntte r b m r r, nd h re mll, one cn lnerzed thee equton nd boundry condton mlrly to [, 7, 5, ]. A reult, ccordng to the equton we recee the followng lner equton nd boundry condton for the mgnetoeltc chrctertc of the deformed body n the form: t u μ M M h m μm j, kuk, j tj, kuk, j ρ.38 t j, j j, j, j j j, e jk hk j j mum k ], b B j kuk j,, j, k,,3.39 [,,, where t σ μ M μ m j j,,, m j j j σ j λδ ju k k μ u j u j, j M j.4,,, t μ The boundry condton cn be wrtten n the form [ Φ ] n [ M tj tj ] ; n [ b ] um, nm[ B ], [ Φ] u X.4 where h [ A] A A jump of the quntty A cro urfce of dcontnuty; grdφ, e, e ; grdφ, e, e nonperturbet mgnetc feld repectely; ;, e Φ nd, e Φ re potentl of perturbed nd M t B B h B h.5μ δ h.4 j j j j j k k k The dependence of b nd m on h cn be obtned n the form [] r r m ˆ h b μ I ˆ h.43 where Î the dentty mtr nd element of the mtr â re defned 6

23 dχ j dχ χ j j d d.44 Then, for the mgnetoeltc med wth nonlner lw of mgnetzton M χ, the moton equton cn be preented n the followng form r r r μ r ρ u u u f ν μ μ t r d [ Iˆ ˆ h] B j, kuk, j e h u ].47 jk[ k, j j, m m, k The component of the ector μ μ f r re gen: μ f μ bh, bh, b3h3,3 b4h, b5h,3 b6h, 3 μ [M j, j j M j, j ] t j, kuk, j.48 μ μ where b χ ; b b 5 3 χ ; b 4 χ 3 ; χ ; b ; b μ Epreon for f μ μ nd f 3 μ μ cn be dered from f μ by cyclc permutton 3. In the boundry condton.4 the relton.4 wll be tken nto ccount lo. From the moton equton nd the boundry condton.4 for the mgnetooft ferromgnetc body wth the lner lw of mgnetzton cn be obtned equton of moton nd boundry condton repectely. 7

24 Notce tht, for the domn outde of the body concdng wth the cuum the equton goernng the mgnetc feld re: curl h e r, d h e.49.5 DIFFEENT MAGNETIC SUSCEPTIBILITIES OF A SOFT FEOMAGNETIC MATEIALS Mgnetc feld, mgnetzton M nd mgnetc nducton B chrcterzng undeformed tte of body re gen : M χ, B μ [ χ ].5 The functon χ μ the mgnetc uceptblty of mterl. In generl, r for mgnetooft mterl wth nonlner dependence Fgure 3 the uceptblty of mterl χ depend only upon the module of mgnetc feld:. Epermentl netgton [5, ] how tht the mgnetc uceptblte χ of dfferent mgneto oft ferromgnetc mterl dependng on the module of lue of the mgnetc feld ntenty cn be ppromted ung the mn cure of mgnetzton wth uffcently hgh precon by the followng formul: Drefou form: χ β / μ rctg α.5 where β B /π, α μ μ / β r 8

25 5 Suceptblty Mgnetc feld Fgure 3: Dependence of mgnetc uceptblty eru mgnetc feld for pure ron ere B denote the nducton turton; μ ntl relte mgnetc permeblty of r the mterl. The equton.5 re ued by [, 8] n the conttute equton of mgnetooft mterl wth nonlner lw of mgnetzton. ylegh form: χ κ b.5 r whch the lner ppromton of the Drefou formul nd pplcble f < c, where c the coercte force [, 3]. 3 Lner dependence: If the coeffcent of nonlnerlyb, then the dependence r χ cont μ.53 r 9

26 cn be obtned from.5. It often ued for wek mgnetc feld n the conttute equton of mgnetooft ferromgnetc mterl wth lner chrctertc [,,3-5,9-3]. 4 Mgneto rgd form mgnetclly turted mterl: χ M /.54 where M B / μ the turton mgnetzton. The numercl lue of coeffcentα, β, M, κ, b r, μ, μ r nd μ r for dfferent ferromgnetc mterl re gen n []. The Eq re content wth the followng Mwell equton nd correpondng condton: Equton n the nternl domn: curl, db.55 B,.56 μ M μ[ χ ] Equton n the eternl domn: e curl e, db.57 B e e e μ M.58 Condton on the urfce of the undeformed body: n [ B ], n [ ].59 Condton t nfnty:, when r /.6 B e B 3

27 3. STESS-STAIN STATE OF FEOMAGNETIC PLANE WIT A MOVING CACK Bed on the equton nd the boundry condton dered n chpter the problem of tre-trn tte of ferromgnetc plne dcued here. To uch purpoe, oft mgnetc ferroeltc body ccountng nonlner lw of mgnetzton nd mmered n mgnetc feld perpendculr to crck lne condered. Aumng tht the procee n mong coordnte re ttonry, Fourer trnform method ued to reduce the med boundry lue problem to the pr of dul ntegrl equton tht re oled nlytclly. The mgnetoeltc tre ntenty fctor obtned nd t dependency upon the crck elocty, mterl contnt nd nonlner lw of mgnetzton preented. 3. MATEMATICAL MODELING A mgnetoeltc plne wth fnte crck of length locted n mgnetc feld,, wth cont. Crck mong wth the contnt elocty V < c where c the peed of propgton of legh we, nd locted n plne long lne, nd Vt < < Vt Fgure 4. In-plne nontrl dplcement re repreented : u u,,, u u,,, u 3. t t 3 where et of mong coordnte ytem, y, z ttched t the center of the mong crck choen uch tht:, t t 3. Vt y, z 3,

28 A bc umpton, the crck propgton occur durng n nterl of tme when n the mong reference ytem of coordnte the mgnetoeltc tte tme-nrnt. y P r - P Fgure 4: Ferromgnetc plne wth crck n mgnetc feld 3. GOVENING EQUATIONS AND BOUNDAY CONDITIONS Goernng Equton The goernng equton of mgnetoeltc med wth nonlner lw of mgnetzton M χ cn be epreed n the form Chpter, Eq..45,.48: Δu u ν j, j μ f μ ρ u μ t 3.3 d [ Iˆ ˆ h] B j, kuk, j 3.4 e h u ] 3.5 jk[ k, j j, m m, k Aumng tht

29 u ϕ, ψ,, u ϕ, ψ,, h grd Φ 3.6 from Eq. 3.3 we cn get the followng two decoupled equton n term of ϕ nd ψ c ϕ ϕ δ Φ c ψ ψ δ Φ 3.7b, t t,, t t, eren, ϕ nd ψ re dplcement potentl functon. In Eq. 3.7b / /, γ λ λ δ, γ c γ λ λ δ c, γ c ν c, ν c μ / ρ, λ χ χ ], [ hc ' ' λ λ{ [ χ χ ]/ χ χ /[ χ ]}/, h c μ / μ Mgnetc potentl Φ determned from γ Φ Φ 3.8,, y y where ' γ χ / χ χ Ung coordnte trnformton 3., the Eq. 3.7b nd 3.8 become: ϕ ϕ r Φ 3.9,, yy, y ψ ψ r Φ 3.9b,, yy, y γ Φ Φ 3.9c,, yy where 3

30 M, M V / c, r δ / c,, The Mch number M <, nce we ume tht the crck propgted t ubonc peed. Boundry Condton On the urfce y the followng boundry condton hould tfy u y,, for >, 3. Φ,, for >, 3.b Φ, d u y,, for <, 3.,, ν ν ν ν u,, u y, y, eφ, y, P mec P mg P, for < 3.b u, u, L Φ, for <, 3. y,, y, where P h χ / ; P / μ nondmenonl mechncl force ctng on the mg c urfce of the crck. mec Notce tht the boundry condton 3. nd 3.b re conequence of ymmetry of the dplcement nd the mgnetc potentl. In Equton , e χ χ χ χ ], d h c [ χ, L hc χ. 4

31 3.3 SOLUTION METODOLOGY Applyng Fourer trnform method to Equton 3.9 the potentl functon re redly obtned ϕ δ α γ C αep[ γ α y]}ep[ α] dα 3.3, y { A αep[ β y] π c α γ β ψ δ α C αep[ γ α y]}ep[ α] dα 3.4, y { B αep[ β y] π c α γ β Φ, y C αep[ γ α y]ep[ α] dα 3.5 π where β α M In Eq A, B nd C re unknown functon to be determned from the boundry condton The unknown functon of new quntty D α : A, B nd C cn be rewrtten dq d γ Q Q L A α D α 3.6 α [ d γ Q Q L ] B α D α 3.7 α C α d D α 3.8 In Eq. 3.6 nd 3.7 5

32 6 c c Q γ γ δ γ γ δ, c c Q γ δ γ γ δ 3.9 It ey to how tht bed on the boundry condton 3.-3., the determnton of unknown α D yeld to the followng dul ntegrl equton: > d D, ] ep[ α α α π 3. < P d D, ] ep[ α α α α π 3. eren, γ γ ν ν ν ν e Q Q d 3. L Q Q d Q d γ 3.b L d Q d Q d γ 3.c 3.4 MAGNETOELASTIC STESSES AND STESS INTENSITY FACTO Inertng Eq. 3.6, nto Eq..4 from Chpter, the mgnetoeltc tree μ μ μ /, /, /, y t y t y t M yy E yy T yy nd μ μ μ /, /, /, y t y t y t M y E y T y we cn epre through α D follow:

33 t T yy, y / μ { ep[ α y] ep[ α y π ] d ep[ γ α y]} α D α ep[ α] dα t T y, y / μ { ep[ α y] ep[ α y π ] where 3 d ep[ γ α y]} α D αep[ α] dα ν ν Q γ Q γ e, 4 Q γ Q L ν ν e hc [ χ χ ] The oluton of dul ntegrl equton my be repreented n form Append B: D α E ep[ α] d 3.5 α where E P 3.6 Subttuton nto Eq ge u fnl repreentton of mgnetoeltc tree: 7

34 8 { /, y P y t T yy π μ d y d y } 3 γ 3.7 [ /, y y P y t T y π μ d y y d y y ] 4 γ γ 3.8 For the pecl ce when y the mgnetoeltc tre μ /, y t T yy he form d d P t T yy } { /, 3 π μ < > d P,,, } { 3 π 3.9 The normlzed tre ntenty fctor cn be epreed } {, lm 3 d P t K c yy I π μ 3.3 or,, M M F M M F K I π 3.3 where 3, d M M M M F, 3.3

35 9, γ γ ν ν ν ν e Q Q d M M M M F o o From Eq. 3.3, when V, e.g. the crck mmoble, we cn get e F e F K V I γ γ 3.34 eren, L Q Q Q Q Q c c F o ~ ~ ~ ~ ~ γ ν ν ν ν γ, 3.35 ~ γ γ δ γ γ δ c c Q, ~ γ δ γ γ δ c c Q. In ce of mgnetooft mterl wth lner lw of mgnetzton cont χ χ from 3.34 drectly follow tht ] [ c I b K ν χ 3.36 where c c h b χ eult 3.36 concde wth the reult dered by [, 9-]. Notce tht, for mgnetooft mterl wth lner lw of mgnetzton, the ntenty fctor I K when / ν χ c b.

36 4. NUMEICAL COMPUTATIONS AND DISCUSSIONS Th Chpter deoted to the numercl dcuon of tre ntenty fctor nd mgnetoeltc tree of oft ferromgnetc plne wth mong crck n trnerl mgnetc fled. The computtonl reult re bed on Eq. 3.3,3.7 nd 3.8. Notce tht for numercl clculton the nonlner lw of mgnetzton re tken n term of nondmenonl prmeter Append C. 4. STESS INTENSITY FACTO Numercl reult he been clculted for the normlzed tre ntenty fctor I K nd n prtculr the dependency of the tre ntenty fctor upon mgnetc feld h c μ / μ nd normlzed elocty of the crck V V / c h been hghlghted. From Eq. 3.3 t poble to conclude tht for I h the ntenty fctor K when V V < c where, V the mmum elocty ld wthn mgneto-eltc umpton, ledng the denomntor of Eq. 3.3 to zero. Thee prmry concluon how tht the ntenty fctor eentlly depend on n eternl mgnetc feld, peed of mong crck nd lo upon phycl prmeter of problem. Further numercl reult 3 3 re crred out for uperpermlloy ν. 5, ρ 8.77 kg m. c / In Fgure 5-8 the effect of normlzed peed of mong crck V V / c on the normlzed ntenty fctor K I for rou lue of normlzed mgnetc feld 9 h re hown. c 3

37 The nlog clculton crred out ung rou lw of mgnetzton n Fgure 5-8. For emple, n Fgure 5, t ble tht the eternl mgnetc feld eentlly chnge the lue I K compre wth purely eltc ce mgnetc feld c h. In ll ce the normlzed ntenty fctor I K eentlly depend upon the encloed mgnetc feld nd the peed of the crck. The numercl clculton how lo: for V / c V <.8, the ntenty fctor I K frt ncree wth ncreng mgnetc feld then hrply decree nd pe through zero, become negte nd fnlly ncree hrply, b when V / c V. 3. e. the peed of the crck cloe to the propgton peed of ylegh we, the ntenty fctor I K wth the ncree of mgnetc feld. Note, tht the lner lw epreed n Eq..53 tke plce t rther wek mgnetc feld. The nonlner lw.5,.5 nd.54 cn be ued for trong mgnetc feld. Synergtc mplcton of the ntercton of mgneto-eltc feld on the ntenty fctor I K for dfferent lw of mgnetzton re dplyed n Fgure

38 DEIFOUS FOM: b / rctg κ / b χ K I Stre Intent y Fcto r Mgnetc Feld Fgure 5: Dependence of K I on dmenonle mgnetc feld for prmeter 6 5 b, κ nd V. K I Stre Intent y Fcto r Mgnetc Feld Fgure 5b: Dependence of I K on dmenonle mgnetc feld for prmeter 6 5 b, κ nd. 8 V 3

39 K I Stre Intent y Fcto r Mgnetc Feld Fgure 5c: Dependence of I K on dmenonle mgnetc feld for prmeter 6 5 b, κ nd. V K I Stre Intent y Fcto r Mgnetc Feld Fgure 5d: Dependence of I K on dmenonle mgnetc feld for prmeter 6 5 b, κ nd. V 33

40 Lner Lw of Mgnetzton: χ κ cont K I Stre Intent y Fcto r Mgnetc Feld I Fgure 6: Dependence of K on dmenonle mgnetc feld for 5 prmeter κ nd V. K I Stre Intent y Fcto r Mgnetc Feld I Fgure 6b: Dependence of K on dmenonle mgnetc feld for 5 prmeter κ nd V. 8 34

41 K I Stre Intent y Fcto r Mgnetc Feld I Fgure 6c: Dependence of K on dmenonle mgnetc feld for 5 prmeter κ nd V. K I Stre Intent y Fcto r Mgnetc Feld I Fgure 6d: Dependence of K on dmenonle mgnetc feld for 5 prmeter κ nd V. 35

42 ylegh form: χ κ b K I Stre Intent y Fcto r Mgnetc Feld I Fgure 7: Dependence of K on dmenonle mgnetc feld for 3 5 prmeter b, κ nd V. r K I Stre Intent y Fcto r Mgnetc Feld I Fgure 7b: Dependence of K on dmenonle mgnetc feld for 3 5 prmeter b, κ nd V. 8 r 36

43 K I Stre Intent y Fcto r Mgnetc Feld I Fgure 7c: Dependence of K on dmenonle mgnetc feld for 3 5 prmeter b, κ nd V. r K I Stre Intent y Fcto r Mgnetc Feld I Fgure 7d: Dependence of K on dmenonle mgnetc feld for 3 5 prmeter b, κ nd V. r 37

44 Mgneto rgd dependence: χ M / K I Stre Intent y Fcto r Mgnetc Feld I Fgure 8: Dependence of K on dmenonle mgnetc feld for 5 prmeter M nd V. K I Stre Intent y Fcto r Mgnetc Feld I Fgure 8b: Dependence of K on dmenonle mgnetc feld for 5 prmeter M nd V. 8 38

45 K I Stre Intent y Fcto r Mgnetc Feld I Fgure 8c: Dependence of K on dmenonle mgnetc feld for 5 prmeter M nd V. K I Stre Intent y Fcto r Mgnetc Feld I Fgure 8d: Dependence of K on dmenonle mgnetc feld for 5 prmeter M nd V. 39

46 4. MAGNETOELASTIC STESSES Mgneto-eltc tree he been clculted by ung Eq Dmenonle T T tree t XY, y t y, y / Pμ nd tyy, y t yy, y / Pμ re hown n Fgure 9- repectely. The tre dtrbuton were eentlly ndependent of the form umed for the mgnetzton lw, o only thee clculted for the Drefou form wll be preented for llutrton purpoe. Fgure 9- re plot of mgnetc nduced tree t XY, y nd t YY, y wth ncreng mgnetc feld. By comprng Fg. 9b wth Fg. 9c, t cn be een, tht t XY, y bruptly chnge gn t dmenonle feld between. 9 nd.. By comprng Fg., b nd c, t YY, y goe from purely negte hed of the crck tp to pote t. 9 to negte gn of.. Tht, for feld of bout. 9 tre round of the crck tp wtche from tenle to compree nd then bck to tenle gn. Th feld trength well beyond the turton feld for the mterl prmeter umed n the clculton. For clculton we umed lo: normlzed elocty of crck: V / c V. ; normlzed dtnce: y /. nd the normlzed prmeter: X / re chnge n nterl [ 3,3]. 4

47 Dmenonle mgnetoeltc tree t XY nd t YY on the y /.urfce; nonlner lw of mgnetzton n Drefou form 6 5 χ b / rctg κ / b for prmeter b, κ ; normlzed crck peed V / c V. t X Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure 9: Dmenonle tre t XY when. t X Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure 9b: Dmenonle tre t XY when.5 4

48 t X Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure 9c: Dmenonle tre t XY when. t X Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure 9d: Dmenonle tre t XY when 4

49 t X Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure 9e: Dmenonle tre t XY when t X Y Mgnetoelt c Stre Mgnetc Feld X / Crck A Fgure 9f: Dmenonle tre t XY when 9 43

50 t Y Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure : Dmenonle tre t YY when. t Y Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure b: Dmenonle tre t YY when.5 44

51 t Y Y Mgnetoelt c Stre Mgnetc Feld X / Crck A Fgure c: Dmenonle tre t YY when.9 t Y Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure d: Dmenonle tre t YY when 45

52 t Y Y Mgnetoelt c Stre X Mgnetc / Crck Feld A Fgure e: Dmenonle tre t YY when t Y Y Mgnetoelt c Stre Mgnetc Feld X / Crck A Fgure f: Dmenonle tre t YY when 9 46

53 CONCLUSIONS In th work the equton of mgnetoeltcty for mgnetooft mterl wth nonlner lw of mgnetzton re preented. Bed on the preent model the tretrn tte of ferromgnetc plne wth mong crck n trnere mgnetc feld h been nlyzed. It h been hown tht the nonlner lw of mgnetzton h qultte nd qunttte nfluence on the mgnetoeltc quntte. Introducng mgnetc feld nto the eltc crck problem mke tre ntenty elocty dependent nd produce ngulrty t elocte cloe to the ylegh elocty. Mgnetc feld ntroduce nomle n the tre dtrbuton round crck. The her tre component chnge gn t crtcl feld nd the norml tre hed of the crck tp wtche from tenle to compree nd bck to tenle gn wth ncree feld trength. The preent model cn be ued n mgnetoeltcty tude where oft ferromgnetc medum re condered nd re under the preence of lrge mgnetc feld. It cn be ueful for mchnng rng ntenty fctor or ncreng toughne mkng ntenty fctor negte. 47

54 POTENTIAL WOK Confrm the nlytcl model wth eperment. To do th, frcture toughne tet cn be performed on n lloy tht h both hgh uceptblty or permeblty nd low ductlty. Cnddte mterl nclude gh Purty Iron nd yperco Fe-49Co-V, reltely brttle ordered lloy wth modertely hgh uceptblty. The model we he deeloped ume homogeneou mgnetc feld wthn the mterl. oweer, demgnetzed ferromgnetc mterl he rndom dtrbuton of mgnetzton drecton rrnged n domn tructure. An mportnt modfcton of the current model would be to ccount for dtrbuton of mgnetzton drecton tht, domn tructure. 48

55 EFEENCES. G.Y. Bgdryn, Vbrton nd Stblty of Mgnetoeltc Sytem n un, Edtorl Bord of Yeren Stte Unerty, Yeren G.Y. Bgdryn, nd D.J. nyn, On Mgnetoeltc Equton of Ferromgnetc Body Wth Nonlner Lw of Mgnetzton n un, Proc. NAS Armen, Phyc, 3, 6, M.V. Belubekyn, nd Yu.M. Khchtryn, On Problem of Mgnetoeltc Stblty of Ferromgnetc Plte n un, Proceedng of Acdemy of Scence of BSS, er. Phy.-Mth. Sc.,, W.F. Brown Jr., Mgnetoeltc Intercton, Sprnger-Verlg, New York M. Bozorth, Ferromgnetm. Toronto-New-York-London Yu. A. Brychko, A.P. Prudnko, Integrl Trnform of Generlzed Functon, Gordon & Brech Sc. Publ., New York, A.C. Erngen, nd G.A. Mugn, Electrodynmc of Contnu Med,,, New York; Sprnger Verlg A.. Gchkech, nd M.J. Solodk, Thermoeltcty Electroconductng Ferromgnetc old, 3-All-Unon Symp, "Theoretcl Problem of Mgnetoeltcty" n un, Yeren, F.D. Gkho, Boundry Vlue Problem I.N. Sneddon, Trn., Addon- Weley, edng, MA, 966, pp D.J. nyn, nd G.T. Phlpoyn, Modelng nd Stblty of Mgnetooft Ferromgnetc Plte n Mgnetc Feld. Proceedng of the oyl Socety London A, UK, 457,

56 . P.M. Kolenko, Introducton to Nonlner Electrodynmc n un, Mnk L.D. Lndu, nd E.M. Lfhtz, Electrodynmc of Contnuou Med. Oford [Englnd]: Butterworth-enemnn F.C. Moon, Mgnetoold Mechnc, Wley, New York F.C. Moon, nd Y.-. Po, Vbrton nd Dynmc Intblty of Bem-Plte n Trnere Mgnetc Feld, Trn. ASME, Journl of Appled Mechnc, 36, G.A. Mugn, Contnuum Mechnc of Electromgnetc Sold, North-ollnd Y. Nhd, Y. Shndo, nd A. Atum, Dffrcton of orzontl Sher We by Mong Interfce Crck, Act Mechnc, 54, Y.-. Po, nd C.S. Yeh, A Lner Theory for Soft Ferromgnetc Eltc Sold, J. Eng. Sc.,, A.D. Polynn, A.V. Mnzhro, ndbook of Integrl Equton, CC Pre,. 9. Y. Shndo, The Lner Mgnetoeltc Problem for Soft Ferromgnetc Eltc Sold wth Fnte Crck, Trn. ASME, Journl of Appled Mechnc, 44, Y. Shndo, A. Oghr, nd K. orguch, Dynmc Bendng of Mgnetclly Sturted Plte wth Thorough Crck n Unform Mgnetc Feld, Theoretcl nd Appled Mechnc, 49,

57 ..F. Terten, Coupled Mgnetomechncl Equton for Mgnetclly Sturted Inultor, J. Mth. Phy., 5, A.A.F. n de Ven, Mgnetoeltc Bucklng of Mgnetclly Sturted Bode, Act Mechnc, 47, X.-J. Zheng, nd X. Wng, Anly of Mgnetoeltc Intercton of ectngulr Ferromgnetc Plte wth Nonlner Mgnetzton, Int. J. of Sold nd Structure, 38,

58 APPENDIX A: NOMENCLATUE - Mgnetc feld chrcterzng undeformed tte of ferromgnetc body M - Mgnetzton chrcterzng undeformed tte of ferromgnetc body B - Mgnetc nducton chrcterzng undeformed tte of ferromgnetc body - Intl ppled mgnetc feld χ - Mgnetc uceptblty of the mterl - Length of fnte crck μ - Mgnetc permeblty of the cuum n - Norml of crck urfce ρ - M denty u - Dplcement component f - Mgnetc olume force μ - Lme coeffcent μr - Mgnetc permeblty c, c - Longtudnl nd trnerl eltc we peed V - Velocty of crck c - Speed of propgton of the legh we M - Mch number h h, h - Induced mgnetc feld ϕ, ψ - Potentl functon chrcterzng dplcement n fed rectngulr coordnte Φ - Mgnetc potentl 5

59 t t t T yy M yy E yy Δ - Lplce opertor T, y, t, y - Mgnetoeltc tree y M, y, t, y - Mwell tre component y E, y, t, y - Pure eltc tree y ν - Poon rto P mec - Mechncl force ctng on the urfce of crck P mg - Mgnetc force P - Sum of the mgnetc nd mechncl force I K - Mgnetoeltc tre ntenty fctor 53

60 APPENDIX B: TE SOLUTION OF DUAL INTEGAL EQUATIONS The pr of dul ntegrl equton of our conderton : π D α ep[ α] dα u y,, > π α D αep[ α ] dα f, < where f known, nd D α to be determned. To obtn the oluton of the pr of equton - we defne the functon E follow:, > α D α ep[ α] dα E 3 E, < Ung the Fourer trnformton theorem, from 3 we cn get: αd α E ep[ α] d π 4 nd ung the epreon 4 cn be rewrtten n the followng form: αdα π E ep[ α] d 5 Notce tht the ntegrl repreentton 5 dentclly tfyng to Eq.. Subttutng 5 nto Eq. nd ung the reult [9, 8]: α ep[ αt] dt, 6 α t we cn get π α ep[ α] α E ep[ α] ddα 54

61 55 α α α α α π d d E ] ep[ f d E d E π π 7 Thu, we wll ole the followng ngulr ntegrl equton: f d E π when < 8 In cle of unbounded functon the oluton of 6 h followng form [9, 8]: τ τ τ τ π d f c E 9 where cont c nd equl to zero becue of condton y y u u d E,,. In our ce cont P f /, thu from 8 we cn get: P E, < 9

62 APPENDIX C: MAGNETIZATION LAWS IN TEMS OF NONDIMENSIONAL PAAMETES Drefou form β μ μ r μ rctg β χ n term of dmenonle prmeter re epreed : κμ μ μhc β μ χ rctg, μ β μ hc μ where κ μ n ntl uceptblty of the mterl; r μ h c μ dmenonle mgnetc feld. For mot ferromgnetc mterl eltc modulu 6 9 μ ~ N / m.from here follow tht h c ~ [ ] ~ m A dmenonle mgnetc feld. After ll, the Drefou form become: 9 β κ μ μ b κ χ rctg rctg μ μ, 9 β b where m A, where 9 b β μμ 9 B B μμ ~ π π 6 56

63 Prmeter b depend on nducton turton B nd for mot ferromgnetc mterl 6 b ~. For uperpermlloy the followng prmeter re : 6 b ~ nd ~ 5 κ. μ r ylegh form χ κ b κ b, r r where κ μ r n ntl uceptblty of the mterl; b r nd b r re nonlner coeffcent nd dmenonle nonlner coeffcent repectely epreed n ylegh form. For uperpermlloy the followng dmenonle prmeter re: 3 5 b r ~ nd κ ~. μ r Mgneto rgd form χ M / M, / where M nd M re turton mgnetzton nd dmenonle turton mgnetzton repectely. 57

64 Vt Stenk rutyunyn w born nd grown up n Armen, one of the mll republc of former Soet Unon. She erned her M.S. Degree n Mechncl Engneerng nd Appled Mthemtc from the Yeren Stte Unerty of Armen n 995. Stenk worked eerch Atnt t the Inttute of Mechnc, where he h been conductng reerch on Pltcty nd trengthenng problem of Cylndrcl nd Concl tube. Currently he purung her Mter /Pd.D. Degree n Mterl Scence nd Engneerng Deprtment t Vrgn Tech nd ded by Dr. W. eynold. er the focue on Mgneto-Eltc Intercton n Ferromgnetc Mterl. 58