A Boundary Integral Formulation of the Plane Problem of Magneto-Elasticity for an Infinite Cylinder in a Transverse Magnetic Field

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1 ngineeing htt://ddoiog/436/eng3545 Published Online Ail 3 (htt://wwwsiog/jounl/eng) A Bound Integl Fomultion of the Plne Poblem of Mgneto-lstiit fo n Infinite linde in Tnsvese Mgneti Field Moustf Sbe Abou-Din Ahmed Foud Ghleb Detment of Mthemtis Fult of Siene io Univesit Giz gt ollege of Siene Ats of Shou Njn Univesit Njn KSA mil: moustf_boudin@hotmilom Reeived August 7 ; evised Mh 3; eted Mh 8 3 oight 3 M S Abou-Din et l This is n oen ess tile distibuted unde the etive ommons Attibution Liense whih emits unestited use distibution eodution in n medium ovided the oiginl wok is oel ited ABSTRAT The objetive of this wok is to esent bound integl fomultion fo the stti line lne stin oblem of unouled mgneto-elstiit fo n infinite mgnetizble linde in tnsvese mgneti field This fomultion llows to obtin nltil solutions in losed fom fo oblems with eltivel simle geometies in ddition to being tiull well-dted to numeil ohes fo moe omlited ses As n lition the fist fundmentl oblem of lstiit fo the iul linde is investigted Kewods: Plne Poblems; Mgnetoelstiit; Tnsvese Mgneti Field; Bound Integl Method Intodution An el vesion of the esent bound integl fomultion ws suggested b one of the uthos (MS Abou-Din) fo the stud of etin oblems in the eletodnmis of uent sheets [] It ws lte on lied fo the solution of genel oblem of nonline gvit wve ogtion in wte [] Due to its effiien the method ws used b the uthos of the esent wok to stud the stti lne stin oblem of the line Theo of lstiit in stesses fo bounded siml onneted egions [3] The themoelsti oblem ws lte on teted long the sme guidelines [4] Reentl the uthos esented bound integl fomultion fo the stti line lne stin oblem of unouled Themomgnetoelstiit fo n infinite linde ing unifom il eleti uent [5] A new eesenttion of the mehnil dislement veto llowed to obtin the omlete solution of the oblem The oosed method elies elusivel on the use of bound integl eesenttions of hmoni funtions is suitble fo both the nltil the numeil tetments of the oblem The numeil set of the oosed fomultion ws ied out b the uthos fo ue lstiit [6] An imlementtion of the method fo boundies with mied geometies ws investigted in [7] In the esent e the fomultion esented in [5] is modified dted to fit the se of n infinite linde of mgnetizble mteil subjet to n etenl tnsvese unifom mgneti field The fist the seond fundmentl oblems of lstiit e teted An lition is given fo the fist fundmentl oblem onl fo iul egion This lition is ment to stess the bilit of the method to hle ses whee nltil solutions e ossible to ovide these solutions eliitel The seond fundmentl oblem m be teted in simil w Poblem Fomultion Bsi qutions Let D be two-dimensionl bounded siml onneted egion eesenting noml oss-setion of the infinite linde ouied b the elsti medium let its bound hve the meti eesenttion s e ssumed ontinuousl s s z () Funtions s diffeentible twie on ee z denote othogonl tesin oodintes in se with oigin O in D unit vetos i j k esetivel Let s be the length s mesued on in the ositive sense ssoited with D fom fied oint Q to genel bound oint Q τ is the unit oight 3 SiRes

2 M S ABOU-DINA A F GALB 395 veto tngent to t Q in the sense of inese of s One hs s s s s τ n () whee the dot ove smbol denotes diffeentition wt s Also k nτ (b) The unknown funtions of the oblem e ssumed to deend solel on the two oodintes qutions of Mgnetoelstiit The genel equtions of stti line Mgnetoelstiit m be found in [5] In wht follows we shll quote these equtions fo non-onduting medi to be used thoughout the tet The ondition fo the etenl mgneti field is inooted oitel qutions of Mgnetosttis ) The field equtions Inside the bod in the bsene of volume eleti hges the field equtions of Mgnetosttis in nononduting medi witten in the SI sstem of units e: ul (3) divb (3b) ul (3) divd (3d) whee is the mgneti field veto B the mgneti indution veto the eleti field veto D the eleti dislement veto The mgneti field ises fom n etenl soue in the fom of n initill unifom mgneti field The equtions of Mgnetosttis e omlemented b: ) The eleti onstitutive eltion D (4) whee is eleti emitivit of the bod ssumed onstnt is the eleti emittivit of vuum with vlue 9 F m 36π 3) The mgneti onstitutive eltions B i j 3 (4b) i ij j whee the indies 3 efe to the z - oodintes esetivel eeted inde denotes summtion ee ij e the omonents of the tenso of the eltive mgneti emebilit of the bod ssumed to deend linel on stin oding to the lw ij ij I ij ij i j 3 (4) whee e onstnts with obvious hsil mening I is the fist invint of the stin tenso with omonents ij ij denote the Koneke delt smbols onstnt efes to the mgneti emebilit of vuum with vlue 7 m ession (4) m be dedued fom genel onstitutive ssumtions but this will be omitted hee An eletil nlogue fo the dieleti tenso omonents unde isotheml onditions m be found elsewhee [8 64 lso 9] We shll ssume qudti deendene of stin on the mgneti field (mgnetostition) Uon substitution of (4) into (4b) one m neglet s n oimtion the thid highe degee tems in the mgneti field omed to the line tem Theefoe B (5) The mgneti veto otentil In view of the geomet of the oblem the mgneti veto otentil hs single non vnishing omonent long the z- is: A A k In view of the oet (3b) of the mgneti indution tking (5) into ount the mgneti field veto m be eesented in the fom ula (6) whee A is the mgneti veto otentil It is usul fo the ske of uniqueness of the solution to imose the ondition diva (6b) Sine we e inteested solel in lne oblems the mgneti field lies in the -lne is indeendent in mgnitude of the thid oodinte z A veto otentil oduing suh field must be of the fom A A k (7) This hoie identill stisfies ondition (6b) whih mens tht funtion A still hs some indetemin In ft it is defined u to n bit dditive onstnt qution (3) edues to fom whih (8) A (9) oight 3 SiRes

3 396 M S ABOU-DINA A F GALB t eh oint of the egion D In the esent qusistti fomultion in view of the ft tht the eleti mgneti fields e unouled thee e no soues fo the eleti field Theefoe () whee efes to fee se suounding the bod In the fee se the equtions of Mgnetosttis hold with ene A () one uses the following deomosition: Funtion A A A A () eesents the modifition of the mg- neti veto otentil in fee se due to the esene of the bod This funtion hs egul behvio t infinit It is suffiient fo the esent uose tht this funtion vnish t infinit t lest s with Funtion A ounts fo the unetubed oiginl onstnt mgneti field If the intensit of this initil field is its dietion is inlined t n ngle to the -is then A os sin (3) The setion of the eession fo A into two ts s in () is of itl imotne fo the numeil tetment of the oblem The equtions of Mgnetosttis e omlemented b the following mgneti bound onditions: ) The ontinuit of the noml omonent of the mgneti indution This edues to the ondition of ontinuit of the veto otentil ie A A son A G (4) b) The ontinuit of the tngentil omonent of the mgneti field (in the bsene of sufe eleti uents) This imlies A A A G son (5) n n n These onditions togethe with the vnishing ondition t infinit of A e suffiient fo the omlete detemintion of the two hmoni funtions A A The mgneti field omonents e eessed s A A A A z (6) whee A denote the hmoni onjugte to A It follows fom (6) tht the funtion A ls the ole of sl mgneti otentil Thus one m invibl oeed with the oblem fomultion using eithe the mgneti sl o the mgneti veto otentil We shll use the ltte The solution of the eletomgneti oblem thus edues to the detemintion of two hmoni funtions A A subjet to the bound onditions (4) (5) qutions of lstiit ) qutions of equilibium In the bsene of bod foes of non-eletomgneti oigin the equtions of mehnil equilibium in the lne ed j ij i j (7) whee ij e the omonents of the totl stess tenso j denotes ovint diffeentition It is woth noting hee tht the totl stess tenso is sometimes deomosed into two ts: mehnil eletomgneti [] in whih se the bound onditions m tke diffeent foms It is well-known tht qution (7) is stisfied if the onl identill non-vnishing stess omonents e defined though the stess funtion U b the eltions U U U (8) ) The onstitutive eltions The genelized ooke s lw m be deived onsistentl fo n oite fom of the fee eneg of the medium using the genel iniles of ontinuum Mehnis It eds [8 see lso 9 fo the eleti nlogue] I ij ij ij i j ij (9) whee i i is the squed mgnitude of the mgneti field In omonents qution (9) gives u v u () oight 3 SiRes

4 M S ABOU-DINA A F GALB 397 u v v (b) u v () whee e Young s modulus Poisson s tio esetivel fo the onsideed elsti medium 3) The kinemtil eltions These e the eltions between the stin tenso omonents ij the dislement veto omonents u i ij iuj jui i j () o in tesin omonents u v (b) u v whee u v st fo u u esetivel 4) The omtibilit ondition The ondition of solvbilit of qution (b) fo u v fo given RS is () These equtions e omlemented with the oe bound onditions to be disussed in detil in subsequent setions qution fo the stess funtion An eqution fo the stess funtion m be obtined fom the genel field equtions witten in ovint fom [] Fo the esent uoses howeve we efe to deive this eqution fo the seil two-dimensionl oblem unde onsidetion Solving () fo the stin omonents using (8) one obtins u U U U (3) v U U U u v U (3b) (3) Substituting fom (3) into () efoming some tnsfomtions using the equtions of Mgnetosttis (3) one finll ives t the following inhomogeneous bihmoni eqution fo the stess funtion U : U The solution of (4) is sought in the fom (4) U U (5) whee e hmoni funtions belonging to the lss of funtions DΙ D D denotes the losue of D suesit denotes the hmoni onjugte Funtion U is n tiul solution of the eqution U (6) m be eessed in the fom of Newton s otentil fte the funtion on the RS hs been detemined It follows fom (5) tht U 4 U P 4 U (7) Using qutions (5) (7) qutions (3b) m be st in the fom whee u U M 4 v U S 4 (8) (8b) oight 3 SiRes

5 398 M S ABOU-DINA A F GALB M (9) S (9b) Funtion A is defined u to n dditive bit onstnt whih m be detemined b fiing the vlue of the funtion t n bitil hosen oint of D Intoduing two new funtions A N R (3) (3b) it n be esil veified using the equtions of Mgnetosttis tht M R N S (3) (3b) qutions (3) iml the eistene of two singlevlued funtions u v in D suh tht u u M N (3) v v R S (3b) with these nottions equtions (8b) tke the fom u U u 4 v U v 4 (33) (33b) It is to be noted tht the ddition of onstnt to the funtion A mounts to dding line tems in to u line tems in to v whih do not lte (33b) A eesenttion fo the mehnil dislement veto omonents Diffeentiting (33) wt integting the esulting eqution wt fte using (3) (3) one gets u U 4 u f f is n bit funtion of (34) whee A simil oedue with (33b) using (3b) (3b) ields v U (34b) v 4 g whee g is n bit funtion of Substituting fom (34b) into (3) we find tht this eqution is identill stisfied if onl if f g fom whih it follows tht both funtions e onstnts theefoe m be eliminted sine thei ontibution eesents igid bod dislement A simil gument holds fo n onstnt dded to the eession fo Fo the following oedue it will be ssumed tht eh one of these two funtions hs been omletel detemined b ssigning to it given vlue t some bitil hosen oint in D Fom (33) (34) one then fom (33b) (34b) b line integtions long n th inside the egion D A joining n bit hosen fied oint M (whih m be bitil hosen in D ) to genel field oint M one obtins whee U u 4 u U v 4 v M u M dn d M M (35) (35b) (35) v RdSd (35d) M the integtion onstnts being bsobed into funtions whih e et to be detemined The mehnil dislement omonents u v given b eessions (35b) e single-vlued funtions in D sine the line integls in (35d) e th indeendent due to eltions (3b) oight 3 SiRes

6 M S ABOU-DINA A F GALB Bound Integl Reesenttion of the Solution The oblem now edues to the detemintion of seven hmoni funtions: AA A (l though the onjugte funtion does not e in the eessions given bove fo the stess dislement funtions it will be equied fo the subsequent nlsis within the oosed bound integl method) We use the well-known integl eesenttion of hmoni funtion f t genel field oint inside the egion D in tems of the bound vlues of the funtion its hmoni onjugte (fte integting b ts enging) s f f s ln R f s ln R d π s n s o in the equivlent fom f f s ln R f s ln R d π n n s (36) (36b) whee is the distne between the field oint in D the uent integtion oint on The hmoni onjugte of (36b) is whee R f f s f s d s π n n s s tn (36) The eesenttion of the onjugte funtion is given b f f s ln R f s ln R d π n s s o in the equivlent fom f π f s ln R f sln R d n n s (37) (37b) when oint tends to bound oint eltions (36) (36b) e esetivel eled b f s f s ln R f s ln R d π s n s (38) f s f s ln R f s ln R d s n n (38b) π If funtion g is defined in the oute egion D is hmoni in this egion vnishes t infinit t lest s with it n be shown tht the integl eesenttion (36 b) is eled b f (39) g s ln R gs ln R d s π n n it being undestood tht the bound vlues g s g s unde the integl sign on the RS e n lulted t oint with mete s on the oute side of When the oint tends to bound oint with mete s then (39) is eled b the integl eltion g s g s ln R g s ln R d s n n (39b) π 3 Solution fo the Mgneti Veto Potentil As noted bove eh of the two funtions A A is defined u to n bit onstnt to be fied b ssigning given vlue to the funtion t n bitil hosen oint in its domin of definition In ode to obtin the bound vlues of the two hmoni funtions A A wite down eqution (38b) fo As equtio n (39b) fo A s then use the b ound onditions (4) (5) to finll get the following integl eqution fo As : As As ln Rd sgs (4) n whee G s π π Gs ln RGsln R d s n (4) (4b) qution (4) is the nonil fom of the well-known line Fedholm integl eqution of the seond kind fo the detemintion of the bound vlues of As ving solved this integl eqution the bound vlues of A s m then be obtined fom (5) Also using oight 3 SiRes

7 4 M S ABOU-DINA A F GALB (4) its solution qution (38b) witten fo As is edued to the following Fedholm integl eqution of the fist kind fo the noml deivtive of this funtion: As ln Rds Gs π As (4) n the solution of whih llows to detemine A on the n bound using the bound ondition (5) Thus the bound vlues of A A s well s of thei noml deivtives m be detemined Finll qutions (36b) (39) ield the vlues of the mgneti veto otentil evewhee in se while qution (36) gives the hmoni onjugte A in the bod 3 Solution fo the Stess Dislement omonents ving obtined the solution fo the mgneti field evewhee in se in the egion D ouied b the mteil we now tun to solve the mehnil oblem fo the stess the dislement omonents in D The stesses e given though the stess funtion U fom eltions (b) these m be ewit ten using eession (5) in tems of the hmoni funtio ns the tiul solution U in the fom U (43) U U fom whih using () one obtins (43b) (43) 4 (44) Thus one the mgneti field hs been uniquel dete mined the deivtive must be uni-vl- ued funtion The mehnil dislement omonents e given fom eltions (35b) whih m ewitten using (5) in tems of the hmoni funtions s u 34 (45) U u v 34 (45b) U v In view of the integl eesenttions (37b) eessions (43) (45) it is suffiient fo the solution of the mehnil oblem to detemine the bound vlues of the hmoni funtions This equies fou indeendent eltions in these un- knowns two of whih e obtined fom eltion (38) witten fo the emining two fom the bound onditions As mtte of ft othe onditions will still be equied to eliminte the ossible igid bod motion Following [5] we fomulte the onditions fo the two following fundmentl oblems: The fist fundmentl oblem whee the stesses e seified on the bound the seond fundmentl oblem whee the dislements e seified on the bound 4 onditions fo the Uniqueness of the Solution 4 onditions fo liminting the Rigid Bod Tnsltion Following [5] 3 4 u T U u 34 U vt v (46) (46b) In tems of the bound vlues of the unknown hmoni funtions ondition (46) beomes 34 s lnr s lnr n s s s + s s ds n R s R U π u T u (47) oight 3 SiRes

8 M S ABOU-DINA A F GALB s ln R s ln R n s s s + s s d s n R s R U π v T v whee R s s (47b) 4 onditions fo liminting the Rigid Bod Rottion This ondition like the fist two is lied onl fo the fist fundmentl oblem We shll equie tht u v o using (45) 4 T h (48) N R whih m be witten in tems of the bound vlues of the unknown hmoni funtions s 4 π N R s s s s ds n R s R (49) 43 Additionl Simlifing onditions We shll equie the following sulement onditions to be stisfied t the oint Q s of the bound in ode to detemine the totlit of the bit integtion onstnts eing thoughout the solution oess These dditionl onditions hve no hsil imlitions on the solution of the oblem Fo detils onening these dditionl onditions the ede is kindl efeed to [3] ) The vnishing of the funtion U its fist ode til deivtives t Q U U U (5) o equivlentl U U U s n (5b) whih in tems of the bound vlues of the unknown hmoni funtions give U (5) U s U n ) The vnishing of the ombintion (5b) (5) (5d) This lst dditionl ondition mounts to detemining the vlue of t Q is hosen fo the unifomit of esenttion s in [5] Let us finll tun to the bound onditions elted to the equtions of lstiit Fo this we onside setel two fundmentl bound-vlue oblems 4 3 The Fist Fundmentl Poblem In this oblem we e given the foe distibution on the bound of the domin D Let f f i f j f τ f n n denote the etenl foe e unit length of the bound Then t genel bound oint Q the stess veto is tken to stisf the ondition of ontinuit σ f n o in omonents n n f n n f (5) The foe f is divided into two ts: f f f (53) e whee f e is the foe of non eletomgneti oigin f is the foe due to the tion of the mgneti field e unit length of the bound The seond foe m be eessed in tems of the Mwellin stess tenso σ s f σ n (54) with σij i j ij (55) Substituting fo in tems of the stess funtion U fo n n tking on- oight 3 SiRes

9 4 M S ABOU-DINA A F GALB ditions (5) into ount the lst two eltions ield U s s f sds Yss s U s f s d s X s s (56) (56b) Using eessions (56) one m esil obtin the tngentil noml deivtive s of the stess funtion U t the bound oint Q U ssy ssx s s (57) U s s Y s s X s n o in tems of the unknown hmoni funtions s s s s s s s s s (58 ) U sy ss X s s s s s s s s s s s s U sy ssx s s n (58b) qutions (58) togethe with eltion (38) witten fo s s : s s ln R s ln R ds π n s (59) s s ln R s ln R d π n s s (59b) fom set of fou intego-diffeentil eltions the solution of whih unde the set of onditions (47) (48) (5) ovides the bound vlues of the unknown hmoni funtions thei hmoni onjugtes The full detemin tion of these funtions inside the domin D (n d hene of the bihmoni t of the stess funtion U) is then hieved b substitution into the qution (37) witten fo The stess funtion U is finl l obti ned b dding u the tiul integl U 43 The Seond Fundmentl Poblem In this oblem we e given the dislement veto on the bound of the domin D Let this veto be denoted d d id j d τ d n n Multiling the estition of eession (45) to the bound b s tht of eession (45b) b s dding one gets 34 ss s s s s s s s U (6) s sds sds s su s sv s Simill if one multilies the estition of eession (45) to the bound b s tht of eession (45b) b s subtting one obtins 34 ss s s s s s s s U s sds sd s n (6b) These lst two eltions m be onvenientl ewitten s 34 ss s s ss s s s (6) U s s d s d s d s T 34 ss s s s s ss s (6b) U s dnsdn s n whee d s dn s e the tngentil noml omonents esetivel lulted t bound oints of the veto the tesin omonents of whih e u v given b equtions (35d) qutions (6b) (o (6b)) togethe with (59b) fom the equied set of simultneous intego-diffeentil equtions fo the detemintion of the bound vlues of the unknown hmoni funtions Φ Ψ thei hmoni onjugtes The full solution of the oblem oeeds s fo the fist fundmentl oblem 44 Ptil Use of the Method In tie if the fom of the bound is simle enough (eg the ile o the ellise) one m ttemt to find nltil foms fo the solution s shown below in the lition oweve fo moe omlited boundies one hs to eu to numeil ohes In this se oight 3 SiRes

10 M S ABOU-DINA A F GALB 43 the diffeentil integl oetos eing in the equtions e to be disetized s usul the oblem of detemintion of the bound vlues of the unknown funtions edues to finding the solution of line sstem of lgebi equtions The full solution inside D is then obtined b numeil integtion of bound integls of the te (36) [f 67] In lte stge if it is equied to detemine bound vlues of some unknown funtions (fo emle the bound dislement fo the fist fundmentl oblem o the bound stesses fo the seond fundmentl oblem) this m be hieved t one if the solution is obtined nltill s in the woked emles Othewise if numeil oh is doted the lultion m oeed b lulting the fist the seond deivtives wt of the equied funtions on the bound in tems of deivtives tken long the bound then substituting these into the oe eessions (fo emle eessions (43) fo the stesses (45) fo the dislements) 5 The iul linde As n illusttion of the oosed sheme we esent hee below the solution of oblem whih n be hled nltill nmel the infinite non-onduting iul elsti linde led in tnsvese onstnt etenl mgneti field Let the noml oss-setion of the linde be bounded b ile of dius enteed t the oigin of oodintes with meti equtions s os s sin π π s whee is the ol ngle in the ssoited ol sstem of oodintes Let iul linde of wek eleti onduting mgnetizble mteil be led in n etenl tnsvesl onstnt mgneti field whih we tke long the -is 5 Solution fo the qutions of Mgnetosttis The solution fo the mgneti veto otentil omonent is obtined following stes simil to those of the eeding setion in the fom: A sin A (6) sin sin (6b) whee is the intensit of the lied mgneti field * The line t in in the eession fo A is just the funtion A in the genel fomultion of the oblem hoosing A to vnish t the oigin one gets A os The oesonding mgneti field omonents e os sin o elted to the sstem of ol oodintes os sin Also u u os s in os (6) (63) (63b) (64) (64b) (64) (65) sin (65b) The bound vlues of the mgneti field outside the bod e used to lulte the Mwellin stess tenso omonents fo the fomultion of the bound ondi tions of elstiit One obtins σ os (66) 5 The lsti Solution σ sin (66b) Tuning now to the detemintion of the stesses dislements one hs U whee we hve intodued the dimensionless mete oight 3 SiRes

11 44 M S ABOU-DINA A F GALB fom whih one obtins with U D (67) D The estitions of the funtion s of thei hmoni onjugtes to the bound e eessed s genel Fouie ensions in the ol ngle of the sstem of ol oodintes Fo t he se unde onsidetion: os os 3 os 3 b sin sin sin 3 Inside the bod: 3 os os d sin sin os 3 os 3 os3 b sin 3 sin 3 sin 3 os os d sin sin The stess funtion U inside the domin D is then U os b sin 3 os os 4 3 os D (68) The fou simlifin g onditions tken t the oint ield 3 D D b 3 d 3 4 Of the two onditions eessing the suession of the igid bod tnsltion one is identill stisfied while the othe gives 3 4 The suession of the igid bod ottion is identill stisfied Thee emins now the bound onditions to be stisfied whih m be siml witten s the onditions of ontinuit of the two stess omonents elted to the sstem of ol oodintes : t * * The stess omonents m be lulted fom U in the ol sst em of oodintes s follows: U U D os os U 63 sin sin U D 3 os os The fist of the elsti bound onditions then gives D while the seond one ields Finll oight 3 SiRes

12 M S ABOU-DINA A F GALB The stess omonents e os (69) (69b) sin os (69) Finll the mehnil dislement omonents e u os u 3 (7) u sin u 3 It is woth noting hee tht the mteil onstnts Nonline Poblem of Fluid Wves to Sstem of Inte- e onl in the eession fo the dil go-diffeentil qutions with n Oenoghil Alition Jounl of omuttionl Alied Mthe- dislement A mesuement of this dislement t the mtis Vol 95 No sufe of the linde ovides the numeil vlue of doi:6/s377-47(98)7-7 the o mbintion The solution fo the elli- [3] M S Abou-Din A F Ghleb On the Bound Intil bound ould ovide two diffeent eltions fo tegl Fomultion of the Plne Theo of lstiit with the detemintion of both Alitions (Anltil Asets) Jounl of omut- 6 onlusion tionl Alied Mthemtis Vol 6 No [4] M S Abou-Din A F Ghleb On the Bound The lne oblem of line unouled Mgnetoels- Integl Fomultion of the Plne Theo of Themoelstiit fo the se of n etenl tnsvesl mgneti tiit (Anltil Asets) Jounl of Theml Stesses field in the bsene of uent hs been tkled using Vol 5 No -9 bound integl fomultion develoed elie b the doi:8/ uthos tested in the simle ses of ue elstiit [5] M S Abou-Din A F Ghleb On the Bound unouled themoelstiit mgneto-themoelstiit Integl Fomultion of the Plne Theo of Themo- Mgin the esene of il ue nt The esented theo netoelstiit Intentionl Jounl of Alied lethe lition onening the iul bound lel tomgnetis Mehnis No oint out t the effiien of the method in oviding nltil solutions wheneve this is ossible RFRNS [6] M S Abou-Din A F Ghleb On the Bound Integl Fomultion of the Plne Poblem of lstiit with Alitions (omuttionl Asets) Jounl of omuttionl Alied Mthemtis Vol 59 No [] M S Abou-Din A A Ashou A Genel Method fo vluting the uent Sstem Its Mgneti Field [7] J A l-sedw et l Imlementtion of Bound of Plne uent Sheet Unifom et fo etin Integl Method fo the Solution of Plne Poblem of Ae of Diffeent Unifom ondutivit with Results fo lstiit with Mied Geomet of the Bound Po Sque Ae Il Nuovo imento Vol No 5 eedings of the 7th Intentionl onfeene on Theoe til Alied Mehnis io - Mh 3 [] M S Abou-Din M A ell Redution fo the [8] L D Lu M Lifshitz L P Pitevskii le- (7b) oight 3 SiRes

13 46 M S ABOU-DINA A F GALB todnmis of ontinuous Medi dition Pegmon Pess Ofod 984 [] K Yun Mgnetothemoelsti Stesses in n Infinitel Long lindil onduto ing Unifoml Distibuted Ail uent Jounl of Alied Sienes Reseh Vol 6 No [9] R J Knos Two-Dimensionl letostition The Qutel Jounl of Mehnis Alied Mthemt- is Vol 6 No [] M M Ad et l Defomtion of n Infinite lliti lindil onduto ing Unifom Ail uent Mehnis of Mteils Vol oight 3 SiRes