JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (JMET)

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1 JOURNAL OF MECHANICAL ENGINEERING AND ECHNOLOGY (JME) Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July-Dcmbr () pp IAEME: JME I A E M E GL MODEL ON PROPAGAION OF SURFACE WAVES IN MAGNEO-HERMOELASIC MAERIALS WIH VOIDS AND INIIAL SRESS S. M. Abo-Dhb Mth. Dpt. Fculty of Scinc if Univrsity Sudi Arbi ABSRAC In this ppr th surfc wvs propgtion in gnrlizd mgnto-thrmolstic mtrils tking (Grn Lindsy) modl with voids nd initil strss is invstigtd. h bsic govrning qutions hv bn formultd in xz-pln nd th mgntic fild is considrd in y-xis tht cts prpndiculr to th wv propgtion. Lms potntil mthod is pplid to solv th problm. h boundry conditions tht th continuity of forcs strsss nd Mxwlls strsss componnts displcmnt componnts ht flux tmprtur nd volum frction fild r stimtd t th intrfcs btwn two dissimilr hlf-spc to obtin th frquncy qution of th surfc wvs in th mdium. Som spcil css with nglcting: (i) th mgntic fild nd initil strss (ii) th mgntic fild initil strss nd voids prmtrs (iii) th mgntic fild initil strss nd thrml prmtrs nd (iv) th mgntic fild initil strss thrml fild nd voids prmtrs r dducd s spcil css from this study. Kywords: Rottion thrmolsticity mgntic fild surfc wvs initil strss voids.. INRODUCION In rcnt yrs mor ttntions hv bn givn to th initil strss on wvs with thrml fild mgntic fild nd voids undr rlxtion tims bcus of its utilitrin spcts of Sismic wvs Erthquks Volcnos nd Acoustics. In th clssicl thory of thrmolsticity whn n lstic solid is subjctd to thrml disturbnc th ffct is flt in loction fr from th sourc instntnously. his implis tht th thrml wv propgts with infinit spd physiclly impossibl rsult. In contrst to convntionl thrmolsticity non-clssicl thoris cm into xistnc during th lst prt of th th cntury. For xmpl Lord nd Shulmn [] by incorporting flux-rt trm into Fourirs lw of ht conduction formultd gnrlizd thory which involvs hyprbolic ht trnsport qution dmitting finit spd for thrml signls. Grn nd Lindsy [] by including tmprtur rt mong th constitutiv vribls dvlopd tmprtur-rt dpndnt thrmolsticity tht dos not violt th clssicl Fourirs lw of ht conduction whn body undr considrtion hs cntr of symmtry nd this thory lso 8

2 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () prdicts finit spd for ht propgtion. Chndrskhrih [] rfrrd to this wvlik thrml disturbnc s scond sound. h Lord nd Shulmn thory of gnrlizd thrmolsticity ws furthr xtndd by Dhliwl nd Shrif [4] to includ th nisotropic cs. A survy rticl on rprsnttiv thoris in th rng of gnrlizd thrmolsticity is du to Htnrski nd Ignczk [5]. h rflction of thrmolstic wvs from th fr surfc of solid hlf-spc nd t th intrfc btwn two smi-infinit mdi in wldd contct in th contxt of gnrlizd thrmolsticity is invstigtd by Sinh nd Sinh [6] Sinh nd Elsibi ([7] [8]). Abd-All nd Al-Dwy [9] studid th rflction phnomn of SV wvs in gnrlizd thrmolstic mdium. Shrm t l. [] invstigtd th problm of thrmolstic wv rflction from th insultd nd isothrml strss-fr s wll s rigid fixd boundris of solid hlf-spc in th contxt of diffrnt thoris of gnrlizd thrmolsticity. hory of linr lstic mtrils with voids is n importnt gnrliztion of th clssicl thory of lsticity. h thory is usd for invstigting vrious typs of gologicl nd biologicl mtrils for which clssicl thory of lsticity is not dqut. h thory of linr lstic mtrils with voids dls th mtrils with distribution of smll pors or voids whr th volum of void is includd mong th kinmtics vribls. h thory rducs to th clssicl thory in th limiting cs of volum of void tnding to zro. Nonlinr thory of lstic mtrils with voids ws dvlopd by Nunzito nd Cowin []. Cowin nd Nunzito [] dvlopd thory of linr lstic mtrils with voids to study mthmticlly th mchnicl bhvior of porous solids. Puri nd Cowin [] studid th bhvior of pln wvs in linr lstic mtril with voids. Isn [4] dvlopd th linr thory of thrmolstic mtrils with voids. Dhliwl nd Wng [5] formultd th ht-flux dpndnt thrmolsticity thory of n lstic mtril with voids. his thory includs th ht-flux mong th constitutiv vribls nd ssums n volution qution for th ht-flux. Cirltt nd Scli [6] dvlopd nonlinr thory of non-simpl thrmolstic mtrils with voids. Cirltt nd Scrptt [7] studid som rsults on thrmolsticity for dilctric mtrils with voids. Mrin [8-] studid uniqunss nd domin of influnc rsults in thrmolstic bodis with voids. Chirit nd Scli [] studid th sptil nd tmporl bhvior in linr thrmolsticity of mtrils with voids. A thory of thrmolstic mtrils with voids nd without nrgy dissiption is dvlopd by Cicco nd Dico []. Cirltt t l. [] prsntd modl for coustic wv propgtion in porous mtril which lso llows for propgtion of thrml displcmnt wv. Singh [4] studid th wv propgtion in homognous isotropic gnrlizd thrmolstic hlf spc with voids in contxt of Lord nd Shulmn thory. Cirltt t l. [5] studid th linr thory of micropolr thrmolsticity for mtrils with voids. Rcntly Aoudi [6] drivd th qutions of th linr thory of thrmolstic diffusion in porous mdi bsd on th concpt of volum frction. Ahmd nd Khn [7] studid th problm of thrmolstic pln wvs in rotting isotropic mdium. Birsn [8] invstigtd th xistnc nd uniqunss of wk solution in th linr thory of lstic shlls with voids. Abd-ll t l. [9] studid th rflction of th gnrlizd mgnto-thrmo-viscolstic pln wvs. Kumr nd Rni [] discussd th dformtion du to moving lods in thrmolstic body with voids. Abd-All t l. [] pointd out mgnto-thrmo-viscolstic intrctions in n unboundd body with sphricl cvity subjctd to priodic loding. Işn [] xplind th nonlinr pln strin of lstic mtrils with voids. 9

3 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () Rcntly Abo-Dhb [] invstigtd th propgtion of P wvs from strss-fr surfc lstic hlf-spc with voids undr thrml rlxtion nd mgntic fild. Singh nd Pl [4] discussd surfc wvs propgtion in gnrlizd thrmolstic mtril with voids. this ppr is motivtd by th linr thory of thrmolsticity with voids dvlopd by Isn [4]. Rcntly Singh nd Pl [4] studid th surfc wv propgtion in gnrlizd thrmolstic mtril with voids which it th spcil cs from this ppr without ffct ll of th mgntic fild nd rottion. In th prsnt ppr th surfc wvs propgtion in gnrlizd mgntothrmolstic mtrils tking GL modl with voids nd initil strss is invstigtd. In Sction th govrning qutions r gnrlizd with th hlp of Grn nd Lindsy thory []. hs qutions r solvd for gnrl solutions. In Sctions 4 nd 5 th prticulr solutions r obtind nd pplid t rquird boundry conditions to obtin th frquncy qution of surfc wvs in thrmolstic mtril with voids. In Sction 6 som limiting css of th problm r discussd nd th rsults obtind r clcultd numriclly nd prsntd grphiclly. In th lst sction som concluding rmrks r givn. NOMENCLAURE α b m ξ r th void mtril prmtrs α t is th cofficint of linr thrml xpnsion β=α t (λ+µ) δ ij is th Kronckr dlt η is th ntropy pr unit mss Θ = Θ / < < λ nd µ r Lms prmtr µ is th mgntic prmbility ρ is th dnsity σ ij r th componnts of strss vctor τ r th componnts of Mxwlls strsss vctor ij τ ndτ r th thrml rlxtions prmtrs Φ is th chng in th volum frction fild χ is th quilibrtd inrti ω is th frquncy B uur is th mgntic induction vctor C is th spcific ht pr unit mss ij r th componnts of strin tnsor E r is th lctric ntnsity vctor F is Lorntz s body forc g is th intrinsic quilibrtd body forcs H uur is th mgntic fild vctor h ur is th prturbd mgntic fild vctor J ur is th lctric intnsity vctor k is th wv numbr K is th thrml conductivity

4 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () m is th thrmo-void cofficint P is th initil strss q i r th componnts of th ht flux vctor S i r th componnts of th quilibrtd strss vctor t is th tim ₀ is th nturl tmprtur of th mdium is th bsolut tmprtur u i r th componnts of th displcmnt vctor c is th phs spd. - GOVERNING EQUAIONS h govrning qutions for n isotropic homognous lstic solid with gnrlizd thrmolsticity with voids nd incrmntl ht flux t rfrnc tmprtur ₀ tking into our ccount GL modl nd th filds (thrml voids nd lstic) r givn s follows σ ij = λ k k β + τ Θ + b Φ P δ ij + µ ij Pw ij t () i.. q + τ q = K Θ () i i S i α i = Φ () ρ η = β + α Θ + m Φ (4) k k g = b ξ Φ + m Θ (5) k k. η = q i i ρ (6) ij = ( u i. j + u j i ) w ij = ( u j. i u i j ). (7) h Mxwll s lctro-mgntic strss tnsor τ ij is givn by h qution of motion ( H h H h ( H h ) ) τ = µ +. δ. (8) ij i j j i k k ij.. ji j + F i ρ u i (9) σ = which tnds to.. P P µ u i. jj λ µ u j ij β τ Θ i + b Φ i + F i = ρ u i. t () h qution of ht conduction undr GL modl ρ C ( Θ + τ Θ ) + β u k k + m ( Φ + τ Φ ) = K Θ ii () α Φ b u ξ Φ + m Θ = ρ χ Φ () i i k k..

5 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () whr F = J B. () Considr tht th mdium is prfct lctric conductor w tk th linrizd Mxwll s qutions govrning th lctromgntic fild tking into ccount bsnc of th displcmnt currnt (SI) s th form h c u rl h = J c u rl E = µ t d iv h = d iv E = (4) whr h = curl u H (5) whr w hv usd H = H + h x z t H = H. (6) ( ) ( ) For two-dimnsionl motion in xz-pln Eqs. ()-() writtn s P u P u λ + µ + µ H + λ µ µ + + H x x z P u Θ Φ u + µ β τ + b = ρ z x x t (7) P u P u λ + µ + µ H + λ µ µ + + H z x z P u Θ Φ u + µ β τ + = ρ x z z t b Θ u u ρ C τ + β + t x t z t Φ Θ Θ + = + t x z m τ K u u α + b + ξ Φ + m Θ = ρ χ Φ Φ Φ () x z x z t whr τ τ = + τ = + τ. t t h displcmnt componnts u nd u my b writtn in trms of th sclr nd th vctor potntil functions φ nd ψ rspctivly s th following form φ ψ φ ψ u = u = +. () x z z x (8) (9)

6 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () Substituting from Eq. () into Eqs. (7)-() w gt.. C φ β τ Θ + b Φ = φ () C b.. C ψ = ψ () S ε Θ = Θ + ε φ + ε Φ... α Φ ξ Φ b φ + m Θ = ρ χ Φ (5) Whr P P λ + µ + µ H µ β = = β = ρ ρ ρ (6) b K β m ε ε ε C S = = = =. ρ ρ C τ ρ C τ ρ C. SOLUION OF HE PROBLEM For th nlyticl solution of Eqs. () (4) nd (5) in th form of th hrmonic trvling wv w suppos tht th solution tks th form φ Θ Φ x z t = φ ( z ) Θ ( z ) Φ ( z ) xp ik ( x ct ). (7) [ ]( ) [ ] [ ] Substituting from Eq. (7) into Eqs. () (4) nd (5) w gt C ( D k ) + ω φ ( z ) τ β Θ ( z ) + b Φ ( z ) = (8) ( ) ( ) φ ( α ξ ρ χ ω ) i ωε ( D k ) φ ( z ) + ε ( D k ) + i ω Θ ( z ) + i ωε Φ ( z ) = (9) b ( D k ) ( z ) + m Θ ( z ) + ( D k ) + Φ ( z ) = () whr d D = ω = k c. () dz Eliminting th constnts φ₁ Θ₁ nd Φ₁ from Eqs. (8)-() w obtin.. (4) whr ( D ) + M ( D ) + N ( D ) + Q = L ()

7 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () L = α ε C M = C ε ( ρχ ω α k²-ξ ) + α ( i ω ε k² ) + α ε ( ω ² C k² ) + i ω α ε τ β + ε b b N C ( iω ε k² ) i ω ε τ β = + ( ρχω α k² ξ ) + ( ω ² C k² ) ε ( ρχ ω α k²-ξ ) α ( i ω -ε k² ) + () + i ω ε ( b m α τ β k ) + ε ( bτ β C m ) Q = ( ω ² C k² ) ( iω ε k² )( ρχω α k²-iω k ε ) i ω ε m τ β ( ρχ ω α k²-iω ε k² ) ) ε b τ β ε b m + + b b k ( w suppos tht m m nd m r th roots of th qution () thn w cn writ th gnrl solutions φ Θ nd Φ in th forms m z ( ) = m z ( ) φ x z t A A A A A A mz m z mz m z m z ik ( x ct ) (4) (5) Φ x z t = η A + η A + η A + η A + η A mz m z mz m z η A 6 m z ik ( x ct ) ( ) Θ x z t = ξ A + ξ A + ξ A + ξ A + ξ A m z m z m z mz m z ξ A 6 m z ik ( x ct ) whr i ωε C ( m n k ) + ω i ωεb ( m n k ) ηn = b ε ( m n k ) iω iωε τ β + + τ βη n C ( m n k ) + ω ξn = n =. b h gnrl solutions ψ of th qution () is gtting s th form whr m 4 z m 4 z ik ( x ct ) ( x z t ) = [ B + B ] (6) (7) φ (8) m ρ = ω (9) µ 4 k. 4

8 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () 4. FORMULAION OF HE PROBLEM W considr pln wvs propgt through th two smi-infinit hlf-spcs of thrmolstic solid with voids which w idntify s th rgion > M λ µ α β b m nd th rgion < z th mdium M [ ] µ α β b m z th mdium [ ] λ s shown in Figur. Mgnto-thrmolstic solid hlf-spc with voids nd initil strss O M x z M For mdium M ( ) ( ) m z m z φ x z t = A ct + A + A (4) m z ( ) x z t A A m z A ik x Φ = + + ct (4) m z ( ) x z t A A m z A ik x Θ = + + ct (4) ψ = (4) ( ) η η η ( ) ξ ξ ξ ( x z t ) m 4 z ik ( x ct ) B. M For mdium ( ) ( ) m z m z m z ik x φ x z t = A ct + A + A ( ) ( ) m z m z m z ik x Φ x z t = η ct A + η A + η A ( ) ( ) m z m z m z ik x Θ x z t = ξ ct A + ξ A + ξ A ψ ( ) Mgnto-thrmolstic solid hlf-spc with voids nd initil strss m z ik ( x ct ) Figur (): Schmtic of th problm 4 (44) (45) (46) x z t = B. (47) 5. BOUNDARY CONDIIONS 5

9 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () h boundry conditions t th surfc tk th following form σ zz + τ zz = σ zz + τ zz σ zx + τ zx = σ zx + τ zx Θ Θ Φ Φ z z z z u = = Θ = Θ Φ = Φ = u u u t z whr K ( ikc τ ) α χ = χ = K ( ikc τ ) α w cn writ qutions (47) in th forms P φ ψ λ + µ H φ + µ + β τ b Θ + Φ = z x z P φ ψ λ + µ H φ + µ + b β τ Θ + Φ z x z φ ψ ψ φ ψ ψ µ + µ µ µ = + x z x z x z x z Θ Θ Φ Φ z z z z φ ψ φ ψ φ ψ φ ψ = + = + x z x z z x z x Θ = Θ Φ = Φ t z =. Substitution from qutions () (8) () nd (7) into th boundry conditions qutions (48) w gt whr = (48) (49) (5) (5) 6

10 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () ( )( ) ( )( ) ( )( ) = λ + µ H m k + µ m τ β ζ + bη = λ + µ H m k + µ m τ β ζ + bη = λ + µ H m k + µ m τ β ζ + bη = i µ k m = i µ k m ( )( ) ( )( ) ( )( ) = λ + µ H m k + µ m τ β ζ + b η 5 = λ + µ H m k + µ m τ β ζ 6 + b η = λ + µ H m k + µ m τ β ζ + b η 7 ( ) ( ) = i µ k m = i µ k m = i µ k m = µ k + m 4 4 = i µ k m = i µ k m = i µ k m = µ k + m = m η = m η = m η = = η 6 η 7 η m m m = m ξ = m ξ = m ξ = = ξ 46 ξ 47 ξ m m m = = = = = = ik = m = m = m = m = m = = ik = 66 = 67 = m m m 7 75 = η = η 7 76 = η = η 7 77 = η = η 74 = = ξ 86 = ξ 87 = ξ. 78 = = ξ = ξ = ξ = = If th initil strss nd mgntic fild r nglctd th rsults obtind r dducd to th rlvnt rsults obtind by Singh nd Pl [4]. 6. NUMERICAL RESULS AND DISCUSSION h following vlus of lstic constnts r considrd Singh [4] for mdiums M nd M rspctivly. 7

11 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () ρ =.7Kg / m λ =.7 Nm µ =.78 Nm Cv =.4 J / kg. K 5 6 α =.688 Nm β =.68 Nm K =.7 Jm dg s 6 5 k =.4 b =.849 m = χ =.75 ξ = ρ =.66 Kg / m λ = 5.65 Nm µ =.46 Nm C v =.787 J / kg. K 9 9 α =.8 Nm β =.9 Nm K =.9 Jm dg s k =.5 b =. m = χ =.6 ξ = Using ths vlus it ws found tht ω =. = 98K. In ordr to gin physicl insight th roots Stonly wv vlocity R( ) nd ttnution cofficint I ( ) hv bn discussd by ssigning numricl vlus to th prmtr ncountrd in th problm in which th numricl rsults r displyd with th grphicl illustrtions. h vritions r shown in Figs. ()-(7) rspctivly. Figs. () nd () show th vritions of th roots of qution () with rspct to th initil strss P with vris vlus of th mgntic fild for th two mdium M nd M. From Fig. () it is ppr tht th roots m m nd m for mdium M hv oscilltory bhvior in th whol rng of th P-xis for diffrnt vlus of th mgnti fild H nd dcrs with n incrsing of th initil strss P. From Fig. () it is clr tht th roots m m nd m for mdium M incrs with n incrsing of P but dcrss with n incrsing of th mgntic fild. It is sn tht for lrg vlus of th mgntic fild mnd m hv n oscilltory bhvior with th vrition of P. Fig. : Vrition of th th bsolut vlus of th roots m m nd m with rspct to initil strss with vris vlus of th mgntic fild 8

12 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () Fig. : Vrition of th th bsolut vlus of th roots m m m with rspct to initil strss with vris vlus of th mgntic fild Fig. 4: Vrition of th Stonly wvs vlocity nd ttnution cofficint with rspct to phs spd c with vris vlus of rlxtion tims with nd without mgntic fild 9

13 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () Fig. 5: Vrition of th Stonly wvs vlocity nd ttnution cofficint with rspct to phs spd c with vris vlus of rlxtion tims with nd without initil strss Fig. 6: Vrition of th Stonly wvs vlocity nd ttnution cofficint with rspct to phs spd c with vris vlus of initil strss with nd without mgntic fild 4

14 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () Fig. 7: Vrition of th Stonly wvs vlocity nd ttnution cofficint with rspct to phs spd c with vris vlus of mgntic fild with nd without initil strss Fig. (4) displys th vrition of th Stonly wvs vlocity nd ttnution cofficint with rspct to phs spd c with vris vlus of rlxtion tims with nd without mgntic fild. It is clr tht Stonly wvs vlocity dcrss with th incrsd vlus of th phs spd c nd with n incrsing of th rlxtion tims but th ttnution cofficint incrss with n incrsing of th rlxtion tims nd th smll vlus of phs spd c nd ftr tht rturn dcrss with th lrg vlus of c lso it is shown tht th Stonly wvs vlocity nd ttnution cofficints tk lrg vlus with th prsnc of th mgntic fild compring with th corrsponding vlus in th bsnc of H. From Fig. (5) it is clr tht th stonly wvs vlocity nd ttnution cofficint dcrs with n incrsing of c lso it is obvious tht Stonly wvs vlocity dcrss with n incrsing of th rlxtion tims in th prsnc of th mgntic fild but incrss with th rlxtion tims vrition in th bsnc of H vic vrs for th ttnution cofficints. Fig. (6) plots th Stonly wvs vlocity nd ttnution cofficint with rspct to phs spd c with vris vlus of initil strss with nd without mgntic fild it is obvious tht Stonly wvs vlocity dcrss with th incrsing vlus of th initil strss P but th ttnution cofficint incrss lso it is sn tht thy tnds to zro s c=5. From Fig. (7) w cn s tht Stonly wvs vlocity nd ttnution cofficint incrs with n incrsing of th mgntic fild H. Finlly it is clr tht Stonly wvs vlocity tks smll vlus in th prsnc of th initil strss P compring with th corrsponding vlus in th bsnc of P but th ttnution cofficint tks th lrgst vlus in th prsnc of P compring with it in th bsnc of P. 7. CONCLUSION It is clr from th prvious rsults tht th ffct of initil strss mgntic fild nd thrml rlxtion tim on vlocity of Stonly wvs nd ttnution cofficint in mgntothrmolstic mtrils with voids w hv th vlus of this wvs incrsd or dcrsd with incrsing of th vlu of mgntic fild initil strss nd thrml rlxtion tim Du 4

15 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () to th complictd ntur of th govrning qutions for th gnrlizd mgntothrmolsticity thory th work don in this fild is unfortuntly limitd in numbr. h mthod usd in this study provids quit succssful in dling with such problms. his mthod givs xct solutions in th lstic mdium without ny ssumd rstrictions on th ctul physicl quntitis tht ppr in th govrning qutions of th problm considrd. Importnt phnomn r obsrvd in ll ths computtions: - h influnc of th initil strss on th roots vi th two mdi M nd M hs n pronouncs ffcts. -h Stonly wvs vlocity nd ttnution cofficints tk lrg vlus with th prsnc of th mgntic fild compring with th corrsponding vlus in th bsnc of H. -Stonly wvs vlocity dcrss with n incrsing of th rlxtion tims in th prsnc of th mgntic fild but incrss with th rlxtion tims vrition in th bsnc of H vic vrs for th ttnution cofficints. 4- Stonly wvs vlocity dcrss with th incrsing vlus of th initil strss P but th ttnution cofficint incrss lso it is sn tht thy tnd to zro s c=5. 5- Stonly wvs vlocity nd ttnution cofficint incrs with n incrsing of th mgntic fild H. 6- Stonly wvs vlocity tks smll vlus in th prsnc of th initil strss P compring with th corrsponding vlus in th bsnc of P but th ttnution cofficint tks th lrgst vlus in th prsnc of P compring with it in th bsnc of P. h rsults prsntd in this ppr should prov usful for rsrchrs in mtril scinc dsignrs of nw mtrils low-tmprtur physicists s wll s for thos working on th dvlopmnt of thory of hyprbolic propgtion of hyprbolic thrmolstic. Rlxtion tim voids nd initil strss xchng with th nvironmnt rising from nd insid nuclr rctors influnc thir nd oprtions. Study of th phnomnon of rlxtion tim nd initil strss nd mgntic fild is lso usd to improv th conditions of oil xtrctions. Finlly it is concludd tht th influnc of mgntic fild initil strss voids prmtrs nd thrml rlxtion tims r vry pronouncd in th surfc wvs propgtion phnomn. h rsults obtind r dducd to th rsults obtind by Singh nd Pl [4] in th bsnc of initil strss nd mgntic fild. ACKNOWLEDGEMENS h uthors r grtful to if Univrsity Sudi Arbi for its finncil support for rsrch numbr 8/4/. 8. REFERENCES [] H. W. Lord Y. Shulmn A gnrlizd dynmicl thory of thrmolsticity J. Mch. Phys. Solid 5 (967) [] A. E. Grn A. Lindsy hrmolsticity J. Elsticity (97) -7. [] D. S. Chndrskhrih hrmolsticity with scond sound: rviw Appl. Mch. Rv. 9 (986)

16 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () [4] R. S. Dhliwl H. H. Shrif Gnrlizd thrmolsticity for nisotropic mdi Qurt. Appl. Mth. (98) -8. [5] R. B. Htnrski J. Ignczk Gnrlizd thrmolsticity J. hrml Strsss (999) [6] A. N. Sinh S. B. Sinh Rflction of thrmolstic wvs t solid hlf-spc with thrml rlxtion J. Phys. Erth (974) [7] S. B. Sinh K. A. Elsibi Rflction of thrmolstic wvs t solid hlf-spc with two thrml rlxtion tims J. hrml Strsss 9 (996) [8] S. B. Sinh K. A. Elsibi Rflction nd rfrction of thrmolstic wvs t n intrfc of two smi-infinit mdi with two thrml rlxtion tims J. hrml Strsss (997)9-46. [9] A. N. Abd-ll A. A. S. Al-dwy h rflction phnomn of SV wvs in gnrlizd thrmolstic mdium Int. J. Mth. Mth. Sci. () [] J. N. Shrm V. Kumr D. Chnd Rflction of gnrlizd thrmolstic wvs from th boundry of hlf-spc J. hrml Strsss 6 () [] J. W. Nunzito S. C. Cowin A Nonlinr hory of Elstic Mtrils with Voids Archiv for Rtionl Mchnics nd Anlysis 7() (979) 75-. [] S. C. Cowin J. W. Nunzito Linr Elstic Mtrils with Voids Journl of Elsticity Vol. No [] P. Puri S. C. Cowin Pln wvs in linr lstic mtrils with voids Journl of Elsticity 5() (985) [4] D. Isn A hory of hrmolstic Mtrils with Voids Act Mchnic 6(-) (986) [5] R. S. Dhliwl J. Wng A ht-flux dpndnt thory of thrmolsticity with voids Act Mchnic (-4) (99) -9. [6] M. Cirltt A. Scli On th nonlinr thory of nonsimpl thrmolstic mtrils with voids Journl of Applid Mthmtics nd Mchnics 7() (99) [7] M. Cirltt E. Scrptt Som Rsults on hrmolsticity for Dilctric Mtrils with Voids Journl of Applid Mthmtics nd Mchnics 75(9) (995) [8] M. Mrin A Uniqunss Rsult for Body with Voids in linr hrmolsticity Rndiconti di Mtmtic 7() (997) -. [9] M. Mrin On th Domin of Influnc in hrmolsticity of Bodis with Voids Archiv dr Mthmtik (4) (997) -8. [] M. Mrin Contributions on uniqunss in thrmolsto-dynmics on bodis with voids Cinc. Mth. (Hvn) 6() (998) -9. [] S. Chirit A. Scli On th Sptil nd mporl Bhvior in Linr hrmolsticity of Mtrils with Voids Journl of hrml Strsss 4(5) () [] S. D. Cicco M. Dico A hory of hrmolstic Mtrils with Voids without Enrgy Dissiption Journl of hrml Strsss 5(5) () [] M. Cirltt B. Strughn V. Zmpoli hrmoporocoustic cclrtion wvs in lstic mtrils with voids without nrgy dissiption Intrntionl Journl of Enginring Scinc 45(9) (7) [4] B. Singh Wv Propgtion in gnrlizd thrmoslstic mtril with voids Applid Mthmtics nd Computtion 89() (7) [5] M. Cirltt M. Svndz L. Buonnno Pln wvs nd vibrtions in th thory of micropolr thrmolsticity for mtril thrmolsticity for mtrils with voids Europn Journl of Mchnics A/Solids 8(4) (9)

17 Journl of Mchnicl Enginring nd chnology (JME) ISSN (Print) ISSN 47-9 (Onlin) Volum Issu July -Dcmbr () [6] M. Aoudi A hory of thrmolstic diffusion mtril with voids Zitschrift für Angwndt Mthmtik und Physik 6( ) () [7] F. Ahmd. A. Khn hrmolstic pln wvs in rotting isotropic mdium Act Mchnic 6 (/4) (999) [8] M. Birsn Existnc nd uniqunss of wk solution in th linr thory of lstic shlls with voids Librts Mth. () [9] A. N. Abd-ll A. A. Yhi S. M. Abo-dhb On th rflction of th gnrlizd mgnto-thrmo-viscolstic pln wvs Chos Solitons Frctls 6() () -. [] R. Kumr L. Rni Dformtion du to moving lods in thrmolstic body with voids Int. J. Appl. Mch. Eng. () (6) [] A. M. Abd-All H. A. Hmmd S. M. Abo-Dhb Mgnto-thrmo-viscolstic intrctions in n unboundd body with sphricl cvity subjctd to priodic loding Appl. Mths. & Comp. 55 (4) [] D. Isn Nonlinr pln strin of lstic mtrils with voids Mths. & Mch. of Solids (4) (6) [] S. M. Abo-Dhb Propgtion of P wvs from strss-fr surfc lstic hlf-spc with voids undr thrml rlxtion nd mgntic fild Applid Mthmticl Modlling 4(7) () [4] B. Singh R. Pl surfc wvs propgtion in gnrlizd thrmolstic mtril with voids Applid Mthmtics ()