THEORY OF HYPERBOLIC TWO-TEMPERATURE GENERALIZED THERMOELASTICITY

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1 Materals Physcs Mechancs 40 (08) 58-7 Receed: February 3, 08 THEORY OF HYPERBOLIC TWO-TEMPERATURE GENERALIZED THERMOELASTICITY Hamdy M. Yussef,, Alaa A. El-Bary 3 Mathematcs Department, Faculty f Educatn, Alexra Unersty, Alexra, EGYPT Engneerng Mechancs Department, Faculty f Engnnerng, Umm Al-Quar Unersty, Makka, KSA 3 Basc Appled Scence Department, Arab Academy f Scence Technlgy, Alexra, EGYPT e-mal: yussefanne005@gmal.cm Abstract. Yussef mpred the generalzed thermelastcty base n tw dstnct temperatures; the cnducte temperature the thermdynamcs temperature whch cncde tgether when the heat supply anshes [, ]. Ths thery has ne paradx, where t ffers an nfnte speed f thermal wae prpagatn. S, ths wrk assumng a new cnsderatn f the tw types f temperature whch depends upn the acceleratn f the cnducte the thermal temperature. Ths wrk ntrduces the prf f the unqueness f the slutn, mreer, ne dmensnal numercal applcatn. Accrdng t the numercal result ths new mdel f thermelastcty ffers fnte speed f thermal wae mechancal wae prpagatn. Keywrds: elastcty, thermelastcty, hyperblc tw-temperature, fnte speed, wae prpagatn. Intrductn Duhamel was the frst t cnsder elastc prblems wth heat changes. Neumann re-dered the equatns btaned by Duhamel. Ths thery f uncupled thermelastcty cnssts f the heat equatn ndependent f mechancal effects, the equatn f mtn cntans the temperature, as a knwn functn. Danlskaya [3] was the frst wh sled a prblem n the cntext f the thery f uncupled thermelastcty wth unfrm heat, t was fr a halfspace subjected t a thermal shck. There are tw defects f ths thery. Ths thery states that the mechancal state f the elastc bdy des nt affect the temperature, whch s nt n accrd wth rght physcal experments. Secnd, the heat equatn beng parablc predcts an nfnte speed f prpagatn fr the temperature, whch agan cntradcts physcal bseratns. Bt [4] ntrduced the cupled thery f thermelastcty n whch the equatns f elastcty heat cnductn became cupled, that agree wth physcal experments, any change f the temperature ges a certan amunt f defrmatn n an elastc bdy ce ersa. The thery f cupled thermelastcty has pred useful fr many prblems. The gernng equatns f ths thery cntan the equatn f mtn, whch s a hyperblc partal dfferental equatn, f the equatn f energy cnseratn, whch s parablc. The nature f the heat equatn mples that f an elastc medum s extendng t nfnty subjected t a thermal r mechancal dsturbance, the effect wll fall nstantaneusly at nfnty, whch cntradcts physcal experments. Hence, a new equatn f energy wth hyperblc type s needed. Lrd Shulman [5] ntrduced the thery f generalzed thermelastcty wth ne 08, Peter the Great St. Petersburg Plytechnc Unersty 08, Insttute f Prblems f Mechancal Engneerng RAS

2 Thery f hyperblc tw-temperature generalzed thermelastcty 59 relaxatn tme fr the partcular case f an strpc bdy. Dhalwal Sheref [6] extended ths thery t nclude the anstrpc case. In ths thery, a mdfed law f heat cnductn ncludng bth the heat flux ts tme derate replaces the cnentnal Furer's law (Cattane's heat cnductn). The heat equatn asscated s hyperblc hence elmnates the paradx f nfnte speeds f prpagatn nherent n bth The secnd generalzatn f the cupled thery f elastcty s the thery f thermelastcty wth tw relaxatn tmes. Müller [7] n a reew f the thermdynamcs f thermelastc slds, suggested an entrpy prductn nequalty, wth the use f whch he cnsdered restrctns n a class f cnsttute equatns. Green Laws [8] prpsed a generalzatn f ths nequalty. Green Lndsay [9] gt an explct ersn f the cnsttute equatns. These equatns were btaned ndependently by Suhub [0]. Ths thery cntans tw parameters that act as relaxatn tmes. The classcal Furer's law f heat cnductn s nt satsfed f the medum under cnsderatn has a center f symmetry. Chen Gurtn [], Chen et al. [, 3] hae cnstructed a thery f heat cnductn n defrmable bdes, whch depends upn tw dfferent temperatures, the cnducte temperature, the thermdynamc temperature. Fr tme-ndependent stuatns, the dfference between these tw temperatures s prprtnal t the heat supply. In the absence f the heat supply, the tw temperatures are dentcal. Fr tme-dependent prblems, hweer, fr wae prpagatn prblems n partcular, the tw temperatures are n general dfferent regardless f the presence f heat supply. The thermdynamc temperature, cnducte temperature, the stran are fund t hae representatns n the frm f a traelng wae plus a respnse, whch happen nstantaneusly thrughut the bdy [4]. Warren Chen [5] nestgated the wae prpagatn n the tw-temperature thery f thermelastcty. Yussef [] ntrduced a new thery f tw-temperature generalzed thermelastcty wth the general unqueness therem fr the bundary mxed ntal alue prblems n ths thery. Yussef cnstructed a new thery f tw-temperature generalzed thermelastcty thery fr the hmgeneus strpc bdy wthut energy dsspatn; he presented the general unqueness thery fr the ntal mxed bundary alue prblems n ths thery [], he dered ts aratnal prncple [6].. Basc Equatns The gernng equatns f an strpc hmgeneus thermelastc medum, as prpsed by Lrd Shulman are [5]: The equatn f mtn σ j, j + F = ρ u, () where σ j s the stress tensr, F s the bdy frce cmpnents, ρ s the densty, u s the dsplacement cmpnents. The cnsttute relatn s: σ = µ e +λe δ γ T T δ, () j j kk j 0 j where γ= ( 3λ+ µ ) α T are the cuplng parameters, T s the dynamcal temperature T 0 beng the reference temperature, e j s the stran tensr λ µ are the elastc cnstants f the materal. The defrmatn ej = ( u, j + uj, ). (3) The nn-furer heat cnductn

3 60 Hamdy M. Yussef, Alaa A. El-Bary q q +t = Kφ, (4), t where φ s the cnducte temperature. Mreer, we hae ρ C T +γ Te = q, (5) E 0 kk, where K s the thermal cnductty, q s the heat flux cmpnents, heat wth cnstant stran. The ncrement f the entrpy η satsfes the fllwng equatns:, 0 CE s the specfc q = ρt η, (6) ρt η= ρc T + T γ e. (7) 0 E 0 j j Equatns (4), (6) (7) frmulate the heat cnductn equatns as prpsed by Yussef [] n the frm: Kφ =, +t ( ρ C T+γTe E 0 kk ), (8) t t φ T= aφ, (9), where a 0s called the tw-temperature parameter, whle,j,k =,,3 are the ndeces fr any general c-rdnates n 3-dmensns. 3. One-Dmensnal Generalzed Thermelastc Half-Space (Classcal Tw- Temperature) Wthut lsng the generalty, we wll cnsder ne-dmensnal strpc hmgeneus thermelastc medum ccupes the half-space x 0, ths medum s at rest n the undefrmed state at zer tme wth unfrm temperature T 0. When t > 0, the bundary x = 0f the half-space subjected t a unfrmly dstrbuted tme-dependent stran temperature, then, the gernng equatns take the fllwng frms: e T e ( λ+ µ ) γ = ρ, x x t (0) φ K = +t ( ρ C T+γTe E 0 ), x t t () φ φ T= a, () x σ= ( λ+ µ ) e γ( T T 0 ), (3) u e =. x (4) The bundary the ntal cndtns are T( x,0) =φ ( x,0) = e( x,0) = T ( x,0) =φ ( x,0) = e ( x,0) = 0, (5) e 0, t = e t, φ 0, t =φ t. (6)

4 Thery f hyperblc tw-temperature generalzed thermelastcty 6 Fr smplcty, use the fllwng nn-dmensnal arables; T T φ T 0 0 σ λ+ µ x,u = c η x,u, t, t = c η t, t, θ=, φ =, σ =, c =, T T λ+ µ ρ ρc E η=. K Hence, e θ e e =, x x t 0 0 (7) (8) φ = +t ( θ+e e ), x t t (9) φ φ θ=β, x (0) σ= e eθ, () γt0 γ where ε =, ε =, β= c λ+ µ η a, ε 0, ε 0, β 0. c η Takng the Laplace transfrm fr the bth sdes f the equatns (8)-() as fllws: ( s) = f ( t) 0 st f e dt. () Hence, de dθ e = s e, (3) dx dx ( s s )( e ) d φ = +τ θ+e, (4) dx d φ φ θ=β, (5) dx σ= e eθ, (6) du e =, dx (7) e 0,s = e s, φ 0,s =φ s. (8) de Elmnatng θ frm equatns (3)-(5), then =α φ+α 3e, (9) dx d φ =α φ+e α e, (30) dx where s+τs αε( βα) s αεε( βα) α =, α = α 3 =. +β ( s+τs ) [ βαεε] [ βαee]

5 6 Hamdy M. Yussef, Alaa A. El-Bary By slng the system n (9) (30), the general slutn wll be as fllws: kx kx e = a φ,e,k,k e + a φ,e,k,k e, (3) φ= b φ,e,k,k kx e + b φ,e,k,k kx e. (3) By usng equatns (5) (3), we btan kx kx k b,e,k,k e θ= b φ + bk b φ,e,k,k e, (33) ( ) ( ) ( ) ( ) where ± k ± k are the rts f the fllwng characterstc equatn k α+α k +αα εαα = 0. (34) By slng the abe algebrac equatn, then ( α +α 3) + ( α +α3) 4 αα3 εαα k =, k ( α +α3) ( α +α3) 4 αα3 εαα =. Fr small alues f tme t, ths crrespnds t a large alue f s Taylr expansn; the fllwng cases wll be dscussed. s by usng 4. One-Temperature Mdel (Lrd-Shulman) T get Lrd-Shulman mdel, β= 0 s, the rts f the characterstc equatn wll be n the fllwng frm s k, = + Q, ( ε, ε, τ ) + O V, s. (35) Mreer, the speeds f the waes are: V, =, (36) +τ +eet ± Ψ where Ψ =εετ + εετ + εετ +τ τ +. (37) Equatn (36) shws that the slutns hae tw waes prpagated wth speed V V ( V < V). V s the speed prpagatn f the mechancal wae V s the speed prpagatn f the thermal wae, the medum has n dsturbance fr whch x > tv wth the fllwng cases: Case (.): τ 0. When τ 0, then, V V as n equatn (36), n ths case, the mechancal the thermal waes prpagate wth fnte speeds whch depend n the materal prpertes, where Lrd Shulman gt that results n the generalzed thermelastcty thery [5]. Case (.): τ = 0. When τ = 0, then, V V, hence, nly the mechancal wae prpagates wth fnte speed, ths speed s cnstant ndependent n the materal prpertes, whle the thermal wae prpagates wth nfnte speed ths case s called cupled thermelastcty r Bt's mdel [4].

6 Thery f hyperblc tw-temperature generalzed thermelastcty 63 Case (.3): ε = 0 τ 0. When ε = 0, then, V V, n ths case, the mechancal wae τ prpagates wth fnte speed that s cnstant desn't depend n the materal prpertes, the thermal wae prpagates wth fnte speed depends n the relaxatn tme nly whch s called uncupled thermelastcty. 5. Classcal Tw-Temperature Mdel T get the classcal tw-temperature thermelastcty mdel f Yussef (5), β 0 hence, the rts f the characterstc equatn f the system n (9) (30) lead t the fllwng cases: Case (.): τ 0 r τ = 0. Fr any alue f τ, then, V +εε V, n ths case, nly the mechancal wae prpagates wth fnte speed depends n the materal prpertes whle the thermal wae prpagates wth nfnte speed. Als, the tw-temperature parameter β des nt affect the speed f the thermal r the mechancal wae prpagatn. Case (.): ε = 0. When ε = 0, then, V V, whch s equalent t the case (.). S, the classcal tw-temperature mdel f thermelastcty Yussef (5) presents nt a perfect mdel, where t generates the nfnte speed f thermal wae prpagatn as the uncupled thermelastcty. 6. Hyperblc Tw-Temperature Generalzed Thermelastcty Thery Accrdng t the results n case (), anther frm fr tw-temperature thermelastcty generates t thermal cnducte heat waes prpagatng wth fnte speed s needed. Nw, defne φ as the acceleratn f the cnducte heat, T as the acceleratn f the dynamcal heat. Assumng the dfference between φ T s the prprtn f the heat supples,.e. φ T = c φ, (38), where c (dstance/tme) s cnstant. Defntn "The cnstant c s equal t the dfference between the acceleratn f the cnducte temperature the acceleratn f the thermal temperature when the heat supply s a unt." T apply the last equatn n the abe ne-dmensnal prblem, we hae t use the dmensnless n (8). Hence we hae φ φ T =β x, (39) c where β = s the dmensnless f the hyperblc tw-temperature parameter. c Then, s k, = + Q, ( ε, ε, τ, β ) + O (40) V, s

7 64 Hamdy M. Yussef, Alaa A. El-Bary V, = +βτ +eeβτ +β τ +e e t +τ ± Ψ, (4) where Ψ = τ + e e t +τ +ε ε τ + ε ε τ β ε ε τ β τ + β τ +β τ. (4) Equatn (4) shws that the slutns hae tw waes prpagatng wth speed V ( V V) < gen by (4), where V V s the speed prpagatn f the mechancal wae, V s the speed prpagatn f the thermal wae, the medum has n dsturbance fr whch x > tv wth the fllwng cases: Case (3.): β 0 τ 0. When β 0 τ 0 then, V V as n equatn (4), n ths case, the mechancal the thermal waes prpagate wth fnte speeds whch depend n the materal prpertes, whch agree wth Lrd Shulman n case (.). Nw, the hyperblc tw-temperature parameter β effects n the speed f the thermal the mechancal wae prpagatn. Case (3.): β 0 τ = 0. When β 0 τ = 0, V V, n ths case, nly the mechancal wae prpagates wth fnte speed ths speed s cnstant ndependent n the materal prpertes, whle the thermal wae prpagates wth nfnte speed, ths case s equalent t cases (.) (.). Case (3.3): β 0, τ 0 ε = 0. When β 0, τ 0 ε = 0, then, V V β +. τ In ths case, the mechancal wae prpagates wth fnte speed that s cnstant desn't depend n the materal prpertes, the thermal wae prpagates wth fnte speed depends n the relaxatn tme the hyperblc tw-temperature parameter, t agrees wth the case (.3). Case (3.4): β = 0 τ 0. When β = 0 τ 0, then, V = V V = V, whch s equalent t the case (.) agree wth Lrd-Shulman results [3]. Case (3.5): β = 0, τ 0 ε = 0. When β = τ 0 0, 0 ε =, then, V V, n ths case, the τ mechancal wae prpagates wth fnte speed s cnstant desn't depend n the materal prpertes, whle the thermal wae prpagates wth fnte speed depends n the relaxatn tme nly. Ths case s equalent t the case (.3) f the uncupled thermelastcty. Case (3.6): β = 0 τ = 0. When β = 0 τ = 0, then, V V, n ths case, nly the mechancal wae prpagates wth fnte speed, ths speed s cnstant ndependent n the materal prpertes, whle the thermal wae prpagates wth nfnte speed ths case s equalent t the case (.) r Bt mdel.

8 Thery f hyperblc tw-temperature generalzed thermelastcty Unqueness Therem Let V be an pen regular regn f space wth bundary S ccuped by the reference cnfguratn f a hmgeneus strpc lnear thermelastc sld. S s assumed clsed bunded. Supplement the equatns f tw temperature-generalzed thermelastcty ()-(8) (38) by prescrbed bundary cndtns []: u = u n S [ 0, ), (43) p = p =σjn j n S S [ 0, ), (44) φ =φ θ =θ n S [ 0, ), (45) where S S superpsed " " dentes the prescrbed alues n arbtrary subsets f S ther cmplements. Als, the ntal cndtns as fllws: u = u, u = u, ϕ=ϕ =θ=θ, φ=θ= 0 n V[0, ) at t = 0 (46) Therem: Gen a regular regn f space V+S wth bundary S then there exsts at mst ne set f sngle-alued functns σ j ( x k, t) ej ( x k, t) wth f C (), u ( x k, t), φ ( x k, t) T ( x k, t) f class C () n V+S, t 0 whch satsfy the equatns ()-(8) (38) the cndtns (43)-(46) where K,C E, λµγ,,,t, ρ,c τ all are pste. Prf: Let there be tw sets f functns (I) (II) (I) (II) (I) (II) σ =σ σ, e = e e, φ=φ φ (I) σj σ, (II) j (I) ej (II) e j etc. let j j j j j j etc. By the lnearty f the prblem, t s clear that these dfferences als satsfy the equatns mentned abe mreer, bth knematc statc bundary cndtns are equal t zer (wth F = Q= 0 ), hmgeneus cunterparts f cndtns (43)-(46), : namely they satsfy the fllwng feld equatns n V ( 0, ) σ = j, j ρ u, (47) σ j = µ ej + ( λekk γθ) δ j, (48) q + τ q = - K φ,, (49) q, = - ρt0η (50) ρt 0η= ρc Eθ+ T 0γ j ej, (5) K φ, = ρce ( θ + τ 0 θ) + γ T0 ( e kk +τ ekk ) (5) φ θ= c φ (53) e = ½ u, ( u ) j, j j, +, (54) wth the bundary the ntal cndtns n (43)-(46). Fr smplcty, the wae par has been mtted. Nw, cnsder the ntegral σ e d = σ u d = σ u d. (59) j j j, j j, j Upn nsertng equatn (47), the latter equatn reduced t e d + ρu u d. (60) ( σj j ) = 0

9 66 Hamdy M. Yussef, Alaa A. El-Bary Usng the equatn (48), hence µ e + λd e γ θd e + ρu u d. (6) [( j j kk j ) j ] = 0 It culd be wrtten as fllws: d ρuu e kk e j e j d e kk d 0 dt λ +µ + γθ =. (6) Substtutng fr e kk frm equatn (5), then d ρuu ρce T0 ekk ej ej d K, d dt λ +µ + + θ T θφ 0 (63) + t ρc θθ d +γ T t θ e d = 0. E 0 kk Frm the well-knwn nequalty q q 0. (64), By usng equatn (49), then K q φ d +τ q q d 0, (65),,, whch ges K q φ d τ qq d 0. (66),,, Insertng equatns (50) (5) n the last equatn, hence K θφ d + τ ρc θθ d + γ T τ θ e d 0. (67), E 0 kk Fnally, frm equatns (63) (67), we btan d ρuu ρce e kk e j e j d 0 dt λ +µ + + θ. (68) T 0 The ntegral n the left-h sde f (68) s ntally zer snce the dfference functns satsfy hmgeneus ntal cndtns. By nequalty (68), hweer, ths ntegral ether decreases (r therefre becmes negate) r remans equal t zer. Snce the ntegral s the sum f squares, nly the latter alternate s pssble, that s ρu u ρc E λ e kk + µ ej ej + + θ d = 0, t 0. (69) T 0 It fllws that the dfferent functns are dentcally zer thrughut the bdy fr all tme ths cmpletes the prf f the therem. 8. Numercal Applcatn T get the numercal result whch ncludes the three mdels f thermelastcty; ne temperature f L-S; classcal tw-temperature mdel the hyperblc tw-temperature mdel, then the ceffcents f the gernng equatns (9) (30) wll be n the frm s+τs αε( Ωα) s αεε ( Ωα) α =, α = α 3 =. (7) +Ω ( s+τs ) [ Ωαεε] [ Ωαee] where 0 fr ne temperature Ω= b fr classcal tw temperature. (73) b / s fr hyperblc tw temperature

10 Thery f hyperblc tw-temperature generalzed thermelastcty 67 Assume the thermal shck prblem as fllws: e 0, t = e t = 0, φ 0, t =φ t =ϕ H t, (74) 0 where H( t) s the Heasde unt step functn ϕ0 s the thermal shck ntensty? By usng Laplace transfrm, then ϕ0 e ( 0,s) = e( s) = 0, φ ( 0,s) =φ ( s) =. (75) s Hence, the slutns n the Laplace transfrm dman n the frms: αϕ kx kx e ( x,s) = e e s k k, (76) ( ) ϕ φ ( x,s) = k α e k α e s k ( ) kx kx 3 3 k ϕ kx ( )( 3) ( )( 3) ( k) kx ( α e ( Ωk )( k α )) e 3 ( ) α e( Ω )( α3) θ x,s = Ωk k α e Ωk k α e s k, (77) k x, (78) ϕ s ( x,s) =, (79) s k k kx ( k k ) e αϕ kx kx u( x,s) = k e ke sk k k k. (80) T nert the Laplace transfrms, we adpt a numercal nersn methd based n a Furer seres expansn [7]. By ths methd, the nerse (t) f s s apprxmated by f f the Laplace transfrm t N e kp kpt f ( t) = f ( ) + Re f + exp, 0 < t < t, t k= t t (8) where N s a suffcently large nteger representng the number f terms n the truncated Furer seres, chsen such that Np Npt exp( t) Re f + exp e, (8) t t where e s a prescrbed small pste number that crrespnds t the degree f accuracy requred. The parameter s a pste free parameter that must be greater than the real part f all the sngulartes f f ( s). The ptmal chce f was btaned accrdng t the crtera descrbed n [7]. The cpper materal was chsen fr purpses f numercal ealuatns, the cnstants f the prblem were taken as fllwng []: 5 K = 386 N/K.sec, α T =.78( 0) K, CE = 383.m / K, η = m / sec 0 0 m =3.86( 0) N / m, =7.76( 0) N / m ε =.6086, ε = , λ, 3 ρ = 8954kg / m, τ = 0.0 sec, T = 93 K, β= 0. β =.0 (Assumed).

11 68 Hamdy M. Yussef, Alaa A. El-Bary Fg.. The cnducte temperature dstrbutn Fg.. The therm-dynamcal temperature dstrbutn

12 Thery f hyperblc tw-temperature generalzed thermelastcty 69 Fg. 3. The stress dstrbutn Fg. 4. The dsplacement dstrbutn

13 70 Hamdy M. Yussef, Alaa A. El-Bary Fg. 5. The stran dstrbutn The Fgures -5 shw the cnducte temperature, the therm-dynamcal temperature, the stress, the dsplacement the stran dstrbutns respectely fr the three mdels; ne temperature mdel, classcal tw-temperature mdel, hyperblc tw-temperature mdel. In Fgures -3, the hyperblc tw-temperature mdel agrees wth ne temperature mdel, they ntrduce fnte speed f the cnducte temperature, the therm-dynamcal temperature, the stress waes prpagatn, whle t s nt n the classcal tw-temperature mdel. In Fgures 4 5, the hyperblc tw-temperature mdel agrees wth ne temperature mdel where the dsplacement the stran waes ansh befre the classcal twtemperature mdel. In Fgure 5, the peak pnts f the stran are clsed n the tw cases f the netemperature mdel the hyperblc tw-temperature mdel, whle the peak pnt f the classcal tw-temperature mdel has a dfferent alue far frm the thers peak pnt. 9. Cnclusn - The classcal tw-temperature generalzed thermelastcty mdel Yussef [] des nt ntrduce fnte speed f the thermal wae prpagatn whch s physcally unacceptable. - Ths wrk ntrduces hyperblc tw-temperature generalzed thermelastcty mdel n whch the thermal wae prpagatn has a fnte speed. 3- The tw-temperature parameter has sgnfcant effects n all the studed felds fr the hyperblc tw-temperature generalzed thermelastcty mdel the classcal twtemperature generalzed thermelastcty mdel. 4- The numercal results f all the studed felds shw that, the therm-mechancal waes f the hyperblc tw-temperature generalzed thermelastcty mdel the netemperature generalzed thermelastcty hae the same atttude. 5- The hyperblc tw-temperature generalzed thermelastcty mdel s a successful mdel t study the behar f the thermelastc materals. Acknwledgements. N external fundng was receed fr ths study.

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