UNIQUENESS, RECIPROCITY THEOREM AND VARIATIONAL PRINCIPLE IN FRACTIONAL ORDER THEORY OF THERMOELASTICITY WITH VOIDS

Transcription

1 Maerals Physcs and Mechancs 6 (3) -4 Receved: November 9, UNIQUENESS, RECIPROCITY THEOREM AND ARIATIONAL PRINCIPLE IN FRACTIONAL ORDER THEORY OF THERMOELASTICITY WITH OIDS Rajneesh Kumar *, andana Gupa Deparmen of Mahemacs, Kurukshera Unversy Kurukshera-369, Haryana, Inda *e-mal: rajneesh_kuk@redffmal.com Absrac. In hs work, a new heory of hermoelascy wh vods s dscussed by usng he mehodology of fraconal calculus. The governng equaons for parcle moon n a homogeneous ansoropc fraconal order hermoelasc medum wh vods are presened. A varaonal prncple, unqueness heorem and recprocy heorem are proved. The plane wave propagaon n orhoropc hermoelasc maeral wh fraconal order dervave and vods s suded. For wo-dmensonal problem here exs quas-longudnal (qp) wave, quas-ransverse (qs) wave, quas-longudnal hermal (qt) wave and a quas-longudnal volume fraconal (q) wave. From he obaned resuls he dfferen characerscs of waves lke phase velocy, aenuaon coeffcen, specfc loss and peneraon deph are compued numercally and presened graphcally.. Inroducon The sudy of dynamc properes of elasc solds s sgnfcan n he ulrasonc nspecon of maerals, vbraons of engneerng srucures, n sesmology, geophyscal and varous oher felds. Such maerals are usually descrbed by equaons of lnear elasc solds, however here are maerals of a more complex mcrosrucure (compose maerals, granular maerals, sols ec.) depc specfc characersc response o appled load. Theory of lnear elasc maerals wh vods s one of he mos mporan generalzaons of he classcal heory of elascy. Ths heory has praccal uly for nvesgang varous ypes of geologcal and bologcal maerals for whch elasc heory s nadequae. Ths heory s concerned wh elasc maerals conssng of a dsrbuon of small pores (vods), n whch he vod volume s ncluded among he knemacs varables and n he lmng case of volume endng o zero, he heory reduces o he classcal heory of elascy. A nonlnear heory of elasc maerals wh vods was developed by Nunzao and Cown []. Laer, Cown and Nunzao [] developed a heory of lnear elasc maerals wh vods for he mahemacal sudy of he mechancal behavor of porous solds. They consdered several applcaons of he lnear heory by nvesgang he response of he maerals o homogeneous deformaons, pure bendng of beams and small ampludes of acousc waves. The small acousc waves n an nfne elasc medum wh vods showed ha wo dsnc ypes of longudnal waves and a ransverse wave can propagae whou affecng he porosy of he maeral and whou aenuaon. The wo ypes of longudnal waves are aenuaed and dspersed; one longudnal wave s assocaed wh elasc propery of he maeral and he second assocaed wh he propery of he change n porosy of he 3, Insue of Problems of Mechancal Engneerng

2 Rajneesh Kumar, andana Gupa maeral. These longudnal acousc waves are boh aenuaed and dspersed due o he change n he maeral porosy. Sngh and Tomar [3] nvesgaed he propagaon of plane waves n an nfne hermoelasc medum wh vods. They found ha hree coupled longudnal waves and one ransverse waves can exs n an nfne hermoelasc medum wh vods. Sngh [4] suded he propagaon of plane waves n a generalzed hermoelasc maeral wh vods n he conex of Lord-Shulman [5] heory. He showed ha here exs hree compressonal and one shear wave n an nfne generalzed hermoelasc maeral wh vods. Carlea and Sraughan [6] presened a heory for acceleraon wave propagaon n an elasc maeral wh vods, whch allows hea ravel a nfne speed. Durng recen years, several neresng models have been developed by usng fraconal calculus o sudy he physcal processes parcularly n he area of hea conducon, dffuson, vscoelascy, mechancs of solds, conrol heory, elecrcy ec. I has been realzed ha he use of fraconal order dervaves and negrals leads o he formulaon of ceran physcal problems whch s more economcal and useful han he classcal approach. There exss many maeral and physcal suaons lke amorphous meda, collods, glassy and porous maerals, manmade and bologcal maerals/polymers, ransen loadng ec., where he classcal hermoelascy based on Fourer ype hea conducon breaks down. In such cases, one needs o use a generalzed hermoelascy heory based on an anomalous hea conducon model nvolvng me fraconal (non neger order) dervaves. Povsenko [7] proposed a quas-sac uncoupled heory of hermoelascy based on he hea conducon equaon wh a me-fraconal dervave of order α. Because he hea conducon equaon n he case nerpolaes he parabolc equaon α = and he wave equaon α=, hs heory nerpolaes a classcal hermoelascy and a hermoelascy whou energy dsspaon. He also obaned he sresses correspondng o he fundamenal soluons of a Cauchy problem for he fraconal hea conducon equaon for onedmensonal and wo-dmensonal cases. Povsenko [8] nvesgaed he nonlocal generalzaons of he Fourer law and hea conducon by usng me and space fraconal dervaves. Youssef [9] proposed a new model of hermoelascy heory n he conex of a new consderaon of hea conducon wh fraconal order and proved he unqueness heorem. Jang and Xu [] obaned a fraconal hea conducon equaon wh a me fraconal dervave n he general orhogonal curvlnear coordnae and also n oher orhogonal coordnae sysem. Povsenko [] nvesgaed he fraconal radal hea conducon n an nfne medum wh a cylndrcal cavy and assocaed hermal sresses. Ezza [] consruced a new model of he magneo-hermoelascy heory n he conex of a new consderaon of hea conducon equaon by usng he Taylor seres expanson of me fraconal order, developed by Jumare [3] as q q KT. Ezza [4] suded he problem of sae space approach o hermoelecrc flud wh fraconal order hea ransfer. The Laplace ransform and sae-space echnques were used o solve a one-dmensonal applcaon for a conducng half space of hermoelecrc elasc maeral. Povsenko [5] nvesgaed he generalzed Caaneo-ype equaons wh me fraconal dervaves and formulaed he heory of hermal sresses. Bswas and Sen [6] proposed a scheme for opmal conrol and a pseudo sae space represenaon for a parcular ype of fraconal dynamcal equaon.

3 Unqueness, recprocy heorem and varaonal prncple... 3 Unqueness, recprocy heorems and varaonal prncple n he heory of hermoelasc maerals wh vods were esablshed by Iesan [7]. Unqueness and recprocy heorems for he equaons of generalzed hermoleasc dffuson problem, n soropc meda, was proved by Sheref e al. [8] on he bass of varaonal prncple equaon. Aouad [9, ] proved he unqueness and recprocy heorems for he equaons of generalzed hermoleasc dffuson problem n boh soropc and ansoropc meda by usng he Laplace ransformaon mehod. Sheref e al. [] nroduced a new model of hermoelascy usng fraconal calculus, proved a unqueness heorem, and derved a recprocy relaon and a varaonal prncple. El-Karamany and Ezza [] nroduced wo models where he fraconal dervaves and negrals are used o modfy he Caaneo hea conducon law [3] and n he conex of wo emperaure hermoelascy heory, unqueness and recprocy heorems are proved, he convoluon prncple s gven and s used o prove a unqueness heorem wh no resrcons mposed on he elascy or hermal conducvy ensors excep symmery condons. Rusu [4] suded he exsence and unqueness n hermoelasc maerals wh vods. Marn [5] presened he conrbuons on unqueness n hermoelasodynamcs for bodes wh vods. Pur and Cown [6] suded he behavor of plane waves n lnear elasc maerals wh vods. Doman of nfluence heorem n he lnear heory of elasc maerals wh vods was dscussed by Dhalwal and Wang [7]. Brsan [8] esablshed exsence and unqueness of weak soluon n he lnear heory of elasc shells wh vods. Sharma [9] suded he exsence of longudnal and ransverse waves n ansoropc hermoelasc meda. Analyss of wave moon a he boundary surface of orhoropc hermoelasc maeral wh vods and soropc elasc half-space was suded by Kumar and Kumar [3]. The curren manuscrp s an aemp o combne hese prevous resuls wh he fraconal order heory of generalzed hermoelascy wh vod. The varaonal prncple s proved and s used o prove he unqueness heorem. Recprocy heorem s also proved. Plane wave propagaon n orhoropc hermoelasc medum s suded. Some specal cases are also deduced.. Basc equaons Followng Iesan [7], he basc governng equaons n ansoropc, homogeneous hermoelasc solds wh fraconal order dervave and vods are: Consuve Relaons: c e B D T () j jlm lm j jk, k j h A D e dat () j, j mn mn g Be d mt (3) j, j, e m a at (4) j, j, q KjT, j (5) Equaons of moon: F u (6) j, j Equaons of equlbraed forces:

4 4 Rajneesh Kumar, andana Gupa h, g l (7) Here,, T=-T s he emperaure change, s he absolue emperaure, T s he reference unform emperaure of he body chosen such ha T / T, s he mass densy, q s he hea conducon vecor, s he equlbraed nera, s he volume dsrbuon funcon whch n he reference sae s, s he volume fracon feld, h are he componens of he equlbraed sress vecor, g s he nrnsc equlbraed force, l s he exrnsc equlbraed force, K j =K j s he hermal conducvy ensor, cjkm cjkm ckmj cjkm cjmk s he ensor of elasc consans, j are j he componens of he sress ensor, u are he componens of he dsplacemen vecor, ej u, j uj, are he componens of he sran ensor, s he enropy per un mass, da maaj Aj j j Djk D jk, and Bj B j are he characersc funcons of he maeral. Fraconal negral and dervave. The Remann-Louvlle fraconal negral s nroduced as a naural generalzaon of he convoluon ype negral (Mller and Ross [3], Podlubny [3], Povsenko [7], Povsenko [8]):. (8) If f d The Laplace ransform rule for hs fraconal negral s LI f Lf. (9) s The Remann-Louvlle fraconal dervave of fraconal order α s defned as he lef-nverse of he fraconal negral I α as n nn d n DRLf DI f n fd n n n d () and for Laplace ransform, he nal values of he fraconal negral dervaves of s of order k=,, 3,..., n- are requred, where I n f and he n RL n k k n k LD f s L f s DI f n n () Followng Ezza [6], he Fourer law for Lord-Shulman (LS) model wh fraconal order dervave s q K jt j,. () The hea conducon equaon for Lord-Shulman (LS) model wh fraconal order dervave n absence of hea sources s

5 Unqueness, recprocy heorem and varaonal prncple... T C T T ju, j m KjT, j (3) where s he hermal relaxaon me, whch wll ensure ha he hea conducon equaon wll predc fne speeds of hea propagaon, C s he specfc hea a he consan sran, α denoes he fraconal order. If he maeral symmery s of a ype ha posses a cener of symmery hen D da are dencally zero. Therefore he sysem of equaons ()-(4) becomes: jk j cjlmelm Bj jt (4) h A j, j (5) g Be mt (6) j, j je, j m at. (7) 3. araonal prncple The prncple of vrual work wh varaon of dsplacemens for elasc deformable body wh vods s: f u u l g d Tu hda ju, j h, d. (8) A On he lef hand sde, we have he vrual work of body forces f, l, g, neral forces u, surface forces T jnjh hn whereas on he rgh hand sde, we have he vrual work of nernal forces. We denoe by n j he ouward un normal of Usng he symmery of sress ensor and he defnon of he sran ensor, he equaon (8) can be wren n he alernave form as f u u l g d Tu h da jej h, d. (9) A 5 Subsung he value of oban j from relaon () and h from relaon () n equaon (9), we f u u l g d Tu hda where A WE B e d Te d () j j j j W c e e de A, d. () jlm j lm j We defne a vecor J (Bo [33]) conneced wh he enropy hrough he relaon

6 6 Rajneesh Kumar, andana Gupa J,. () Usng Eqs.(3), (7) and (), we oban d d TL j J T, j d d (3) J e m at (4), j, j where L j s he ressvy marx and s he nverse of he hermal conducvy Kj and we used he noaon C at Mulplyng boh sdes of Eq. (3) by J and negrang over he regon of he body and usng he dvergence heorem wh he ad of (4), we oban TJ j nda j jted j mt d F H (5) A where a (6) F T d F a T Td T dj d J dj d J H L JdH T L Jd d d d d j j j j (7) Subsung he value of g from (3) n relaon (), we oban f u u l d Tu hda m Td Te d j j A where W E R G (8) R B jej d G d. (9) Elmnang negrals jtejd and varaonal prncple n he followng form: m Td from Eqs. (5) and (8), we oban he W E R G H F f u ud l d TudA hda TJndA (3) A A A On he rgh-hand sde of Eq. (3), we fnd all he causes, he mass forces, neral forces, he surface forces, he heang poenal and equlbraed sress vecor on he surface A boundng he body.

7 Unqueness, recprocy heorem and varaonal prncple Unqueness heorem We assume ha he vrual dsplacemens u, he vrual ncremen of he emperaure T, ec. correspond o he ncremens occurrng n he body. Then u T u d ud T d Td ec (3) and Eq. (3) wh he ad of (3) reduces o he followng form: W E R G H F f u ud l d TudA h da TJndA (3) A A A Now uud (33) D d (34) where uud and D d volume. We also have are he knec energy of he body enclosed by he F E G T A d j, (35) Makng use of (33), (34) and (35) n Eq. (3), we oban d W R H D T Aj, dd d fud ld TudA hda TJndA (36) A A A The above equaon s he bass for he proof of he followng unqueness heorem. Theorem. There s only one soluon of he equaons (6), (7) and (3), subjec o boundary condons on he surface A T jnj T h hn j h u u T T, and he nal condons a = u u u u T T T T where h T u u T T are known funcons.

8 8 Rajneesh Kumar, andana Gupa We assume ha he maeral parameers sasfy he nequales T CE m a (37) and cjlmljajbjjdjk are posve defne. Proof. Le u T and u T be wo soluon ses of equaons (6), (7), ()-(7). Le us ake u u u T T T. (38) The funcons u, T and sasfy he governng equaons wh zero body forces and homogeneous nal and boundary condons. Thus hese funcons sasfy an equaon smlar o he equaon (36) wh zero rgh hand sde, ha s d W R H D T A d d d j, (39) Snce we have Lj L from equaon (7), we oban j dh d T d J T L jjjjd L jj d d d d (4) Usng (4) n (39), we oban T d J j j j j d W R D T A d L J d T L JJd. (4), d d Usng nequales (37) n (4), we oban d T d J W R D T Aj d LjJ d. (4), d d We hus see ha he expresson T d J W R D T A d L J d (43), d j j s a decreasng funcon of me. We also noe ha he expresson T Aj d, occurrng n he expresson (43) s always posve. Thus he expresson (43) vanshes for =, due o he homogeneous nal condons, and mus be always non-posve for >. Usng nequaly (37), follows mmedaely ha he expresson (43) mus be dencally zero for >. We hus have u T ej j Ths proves he unqueness of he soluon o he complee sysem of feld equaons subjeced o he nal and boundary condons.

9 Unqueness, recprocy heorem and varaonal prncple Recprocy heorem We shall consder a homogeneous ansoropc fraconal order generalzed hermoelasc body wh vods occupyng he regon and bounded by he surface A. We assume ha he sresses j and he srans e j are connuous ogeher wh her frs dervaves whereas he dsplacemens u, emperaure T and he volume fracon feld are connuous and have connuous dervaves up o he second order, for x A The componens of surface racon, normal componen of he hea flux and he normal componen of equlbraed sress vecor a regular pons of, are gven by T nq K Tnh hn j j j, j (44) respecvely. We denoe by n j he ouward un normal of To he sysem of feld equaon, we mus adjon boundary condons and nal condons. We consder he followng boundary condons: u x U x T x x x x (45) for all x A ; and he homogeneous nal condons u x u x T x T x x x (46) for all x We derve he dynamc recprocy relaonshp for a fraconal order generalzed hermoelasc body, whch sasfes Eqs. (6), (7), ()-(7), he boundary condons (45) and he homogeneous nal condons (46), and subjeced o he acon of body forces F x surface racon h x, he volume fracon feld x and he hea flux qx. Performng he Laplace negral ransform defned as s f xs Lfx fxe d (47) on Eqs. (6), (7), ()-(7) and omng he bars for smplcy, we oban j cjlmelm Bj jt (48) h A j, j (49) g Be mt (5) j j, j e m at (5), j F s u (5) j, j q s KjT, j (53)

10 Rajneesh Kumar, andana Gupa C s s T T s s ju, j m KjT, j h, g l s. (55) We now consder wo problems where appled body forces, volume fracon feld and he surface emperaure are specfed dfferenly. Le he varables nvolved n hese wo problems be dsngushed by superscrps n parenheses. Thus, we have u e T for he frs problem and u e T for he second j j j j problem. Each se of varables sasfes he sysem of equaons (48)-(55). Usng he sran-dsplacemen relaon, he assumpon j j and he dvergence heorem, wh he ad of (44) and (5), we oban e d T u da s u u d F u d. (56) j j A (54) A smlar expresson s obaned for he negral Eq. (56), follows ha j j e d, from whch ogeher wh e e d T u T u da F u F u d. (57) j j j j A Now mulplyng equaon (48) by j j e and e for he frs and second problem respecvely, Subracng and negrang over he regon and usng he symmery properes of c jlm, we oban j j j j j j j j j j e e d B e e d T e T e d. (58) Equang Eqs. (57) and (58), we ge he frs par of he recprocy heorem A T u T u da F u F u d j j j j j j B e e d T e T e d (59) whch conans he mechancal causes of moon FT From equaon (54), we oban C s s T T s s ju, j m KjT, j (6) Now mulplyng equaon (6) by T and T for he frs and second problem respecvely, subracng and negrang over he regon, we oban j, j, j, K T T T T d

11 Unqueness, recprocy heorem and varaonal prncple... T s s e, T e, T d T s s m T T d. (6) j j j Usng dvergence heorem and wh he ad of (44) and (45), equaon (6) becomes q q d T s s e T e T d T s s m T T d (6) j j j Equaon (6) consues he second par of he recprocy heorem whch conans he hermal causes of moon and q. Usng equaon (49) and (5) n (55), we oban A j, Bje, j mt l s. (63) Now mulplyng equaon (63) by () and () for he frs and second problem respecvely, subracng and negrang over he regon, we oban A j,, d B j e j e j d m T T d l,,. (64) Usng dvergence heorem and wh he ad of (5), (44) and (45), Eq. (64) becomes A h h da j j j B e e d m T T d l d (65) The equaon (65) consues he hrd par of recprocy heorem whch conans volume fracon feld. Elmnang he negralsbj e, j e, j dand j e, j T e, j T d from equaons (59), (6) and (65), we oban T s s T u T u da T s s F u F u d A T s s h h da T s s l d q q d A (66) Ths s he general recprocy heorem n he Laplace ransform doman. To nver he Laplace ransform n he eqs.(59), (6), (65) and (66) we shall use he convoluon heorem

12 Rajneesh Kumar, andana Gupa L F s G s f g d g f d (67) and he symbolc noaons f x Mf. (68) Inverng Eq. (59), we oban he frs par of he recprocy heorem n he fnal form A T x u x dda F x u x dd B x e x dd T x e x dd S. (69) j j j j Here S ndcaes he same expresson as on he lef-hand sde excep ha he superscrps () and () are nerchanged. Inverng Eq. (6), we oban he second par of recprocy heorem n he fnal form A j j Me q x x dda T T x x dd M T mt x x dd S. (7) Inverng Eq.(65), we oban he hrd par of recprocy heorem n he fnal form A j j h x x dda Be x x dd m T x xdd l xd S (7) Fnally, nverng Eq.(66), we oban he general recprocy heorem n he fnal form Mu Mu A T T x x dda T F x x dd M A T h x x dda T l x d q x x dda S. (7) A

13 Unqueness, recprocy heorem and varaonal prncple Orhoropc meda The equaons (6),(7) and (3) wh he ad of (4)-(6) whou exrnsc equlbraed force, body force and hea sources for fraconal order generalzed hermoelasc orhoropc medum wh vods are, c u c c u c u c c u c u B T u (73), 66, 66, , 3 55, 33,, c u c u c c u c c u c u B T u (74), 66, 66, , 3 44, 33,, c u c u c u c c u c c u B T u (75) 55 3, , 33 3, , , 3 3, 3 3, 3 3 mt A A A u B u B u B (76) KT KT KT,, 3, 33,,, 33 3,, 3, 3 3 C T T u, u, u3, 3 3 m (77) where Aj A j Bj B j j j Kj K j s no summed. In he above equaons (73)-(77), we use he conracng subscrp noaons,, 333, 34, 35, 6 o relae c jkm o c ln (,j,k,m=,,3 and l,n=,,3,4,5,6). Now we wll dscuss wo-dmensonal plane wave propagaon n homogeneous, orhoropc generalzed hermoelasc medum wh vods. For wo dmensonal problem, we have u u u x x T x x. (78) We defne he followng dmensonless quanes x u T x u T c c c c (79) where c c C c K. Upon nroducng he dmensonless quanes defned by equaon (79) n equaons (73)-(77) and wh he ad of (78) and afer suppressng he prmes, we oban u u u T u (8), 33, 33, 3,, u u u (8), 33, u u u T u (8) 3, 4 3, 33, 3 5, 3, 3 3 u u T (83) 7, 8, 33 9, 3, 3 3 T 4 u, u3, 3 5 T, KT, 33, (84)

14 4 Rajneesh Kumar, andana Gupa where are gven n Appendx. The equaon (8) corresponds o purely quas-ransverse wave mode ha decouples from he res of he moon and s no affeced by he vods and hermal. 7. Soluon of he problem For plane harmonc waves, we assume he soluon of equaons (8)-(84) of he form u u u T U U U T exp x l x l (85) where s he crcular frequency and s he complex wave number. U, U, U 3, T * and * are undeermned amplude vecors ha are ndependen of me and coordnaes x (=, 3), l and l 3 are he drecon cosnes of he wave normal ono he x -x 3 plane wh he propery l l3 Upon usng soluons (85) n he equaons (8),(8)-(84), we oban l l3 U ll 3U3 ll 33 lt (86) ll U l l U l l T (87) l U l3 U3 l 8 l 39 7 T (88) l U l U l l K T (89) 3 where are gven n Appendx. The non-rval soluon of he sysem of equaons (86)-(89) s ensured by a deermnanal equaon gven by l l ll ll l ll l l l l l l l l l l l l K The equaon (9) yelds o followng polynomal characersc equaon n as A B C D E, (9) where he coeffcens A, B, C, D, E and K are gven n Appendx. Solvng equaon (9), we oban egh roos of, ha s, 3 and 4. Correspondng o hese roos, here exs four waves n descendng order of her veloces, namely a quas-longudnal (qp) wave, quas-ransverse qs wave, quas-longudnal hermal (qt) wave and a quaslongudnal volume fraconal (q) wave. Now we derve he expressons of phase velocy, aenuaon coeffcen, specfc loss and peneraon deph of hese ypes of waves as:. (9)

15 Unqueness, recprocy heorem and varaonal prncple... 5 Phase velocy. The phase velocy s gven by 34, (9) Re where 34 are, respecvely, he veloces of qp, qs, qt and q waves. Aenuaon coeffcen. The aenuaon coeffcen s defned as Q Im 34, (93) where Q 34 are, respecvely, he aenuaon coeffcens of qp, qs, qt and q waves. Specfc loss. The specfc loss s he rao of energy W dsspaed n akng a specmen hrough a sress cycle, o he elasc energy (W) sored n he specmen when he sran s a maxmum. The specfc loss s he mos drec mehod of defnng nernal frcon for a maeral. For a snusodal plane wave of small amplude, Kolksy [34] shows ha he specfc loss W / W equals 4 mes he absolue value of he magnary par of o ha of real par of.e. W Im R 4 34 W Re Peneraon deph. The Peneraon deph s defned by. (94) S 34. (95) Im Parcular cases. () In he absence of vod effec.e. when m A A3 B B3 he characersc equaon (9) reduces o he characersc equaon correspondng o he orhoropc fraconal order generalzed hermoelasc medum: 6 4 A B C D where A B C and D are gven n Appendx. () If α= n equaon (),(3) and (89), he correspondng resuls reduce o he case of Lord-Shulman heory of generalzed hermoelascy wh vods. 8. Numercal resuls and dscusson In hs secon, numercal dscusson for phase veloces, aenuaon coeffcen, specfc loss and peneraon deph of quas-longudnal (qp) wave, quas-ransverse waves (qs), quaslongudnal hermal (qt) wave, quas-longudnal volume fraconal (q) wave and a s presened. The maeral chosen for hs purpose s Cobal, whose physcal daa are gven by [35]: c 37 N / m c 65 N / m c3 7 N / m c 358 N / m c 79 N/ m T 98K 4 s Kg / m

16 6 Rajneesh Kumar, andana Gupa 7 4 N / m deg N / m deg C 4 7 J/Kg deg, 5 6 K 69 W/m deg K3 69 W/m deg, 88 od parameers are m A N A N 3 4 N / m, B N B3 7 4 N m N / mk We can solve equaon (9) wh he help of he sofware Malab 7..4 and afer solvng he equaon (9) and usng he formulas gven by (9)-(95), we can compue he values of phase velocy, aenuaon coeffcen, specfc loss and peneraon deph for nermedae values of frequency ( ) and dfferen values of fraconal order dervave.e. α =.5,.,.5. In all he fgures horzonal lnes, square boxes and vercal lnes corresponds o =.5,., and.5 respecvely. Phase velocy. Fgure shows ha for all fraconal orders, he values of ncrease smoohly wh ncrease n values of. On comparng he values of for dfferen fraconal orders, he values of ncrease wh ncrease n fraconal order. I s evden from Fgure ha frsly he values of decrease smoohly bu lasly reman consan..4.3 Phase elocy ( ) ncy e u q..5 e Fr Fg.. araon of phase velocy w.r.. frequency Phase elocy ( ) ncy Freque Fg.. araon of phase velocy w.r.. frequency.

17 7 Unqueness, recprocy heorem and varaonal prncple... Fgure 3 shows ha he values of 3 frs ncrease rapdly and hen decrease smoohly and shows he consan behavor. Fgure 4 ndcaes ha values of 4 frs decrease rapdly and fnally consan behavor s noced Phase elocy ( 3 ) ency Frequ Fg. 3. araon of phase velocy 3 w.r.. frequency. 5 3 Phase elocy ( 4 ) ency Frequ Fg. 4. araon of phase velocy 4 w.r.. frequency. Aenuaon Coeffcen. I s noced from Fg. 5 ha values of Q ncrease smoohly wh ncrease n values of. The values of Q ncrease wh ncrease n fraconal order α. Fgure 6 shows ha Q frs decrease and hen ncrease smoohly wh ncrease n and fnally becomes consan. From Fg. 7, s evden ha Q3 ncrease wh ncrease n. Fgure 8 ndcaes ha Q4 shows he behavor oppose o ha of Q3. Specfc loss. Fgure 9 shows ha for α=.5, R decrease smoohly for 3 and becomes consan for 3 whereas for α=., decrease rapdly and for α=.5, decrease slowly. The values of R ncrease wh ncrease n fraconal order α. Fgure ndcaes ha R frs decrease rapdly and lasly becomes consan. The values of R decrease wh ncrease n fraconal order α and maxmum value occurs for α=.5. I s evden from Fg. ha he values of R3 ncrease wh ncrease n and lasly shows he consan behavor. Fgure shows ha he behavor of he values of R4 s same as ha of R3.

18 8 Rajneesh Kumar, andana Gupa Aenuaon Coeffcen ( Q ) c n y Freque Fg. 5. araon of aenuaon coeffcen Q w.r.. frequency..3 Aenuaon Coeffcen ( Q ) ncy Freque Fg. 6. araon of aenuaon coeffcen Q w.r.. frequency..4 Aenuaon Coeffcen ( Q3 ) ncy Freque Fg. 7. araon of aenuaon coeffcen Q3 w.r.. frequency.

19 Aenuaon Coeffcen ( Q4 ) Unqueness, recprocy heorem and varaonal prncple cy Frequen Fg. 8. araon of aenuaon coeffcen Q4 w.r.. frequency..5.5 Specfc Loss ( R ) ncy Freque Fg. 9. araon of specfc loss R w.r.. frequency Specfc Loss ( R ) ncy Freque Fg.. araon of specfc loss R w.r.. frequency.

20 Rajneesh Kumar, andana Gupa.4..8 Specfc Loss ( R3 ) ncy..5 Freque Fg.. araon of specfc loss R3 w.r.. frequency Specfc Loss ( R4 ) cy n Freque Fg.. araon of specfc loss R4 w.r.. frequency..4.3 Peneraon Deph ( S ) cy Frequen Fg. 3. araon of peneraon deph S w.r.. frequency.

21 Peneraon Deph ( S ) Unqueness, recprocy heorem and varaonal prncple ency Frequ Fg. 4. araon of peneraon deph S w.r.. frequency. 5 7 Peneraon Deph ( S3 ) ency Frequ Fg. 5. araon of peneraon deph S3 w.r.. frequency Peneraon Deph ( S4 ) ency Frequ Fg. 6. araon of peneraon deph S4 w.r.. frequency.

22 Rajneesh Kumar, andana Gupa Peneraon deph. Fgure 3 shows ha peneraon deph S decrease smoohly wh ncrease n values of. Fgure 4 ndcaes ha S ncrease rapdly for small values of and fnally consan behavor s noced. The values of S ncrease wh ncrease n fraconal order. I s evden from Fg. 5, ha S 3 rapdly ncrease and decrease nally and fnally becomes consan. The values of S 3 ncrease wh ncrease n fraconal order. Fgure 6 depcs ha he behavor of he values of S 4 s oppose o ha of 4 n Fg. 4. The values of peneraon deph are magnfed by mulplyng S, S 3, and S 4 by 3,, and 3 respecvely. 9. Conclusons A model of ansoropc, homogeneous hermoelasc solds wh fraconal order dervave and vods based on he heory of Lord and Shulman s gven. Usng he varaonal heorem, he unqueness heorem of soluon of he nal boundary value problem s proved, and he dynamc recprocy heorem s derved for he gven model. The governng equaons for he orhoropc hermoelasc maeral wh fraconal order dervave and vods are presened. For wo dmensonal problem here exs quaslongudnal wave (qp), quas-ransverse waves (qs), quas-longudnal hermal wave (qt) and a quas-longudnal volume fraconal wave (q). The phase velocy, aenuaon coeffcen, specfc loss and peneraon deph are compued numercally and presened graphcally. Some parcular cases are also dscussed. From Fgures s observed ha values of phase velocy ncrease, aenuaon coeffcens Q and Q decrease, specfc loss R ncrease, specfc loss R decrease, peneraon deph S and S 3 ncrease wh ncrease n he fraconal order. Appendx c c c c c B c B c c c c c A A B B m C c c c c c K T Tmc 3 K3 4 5 K 3 K K K A = R R 7 R 3 R 9 + R 3 R 6 R 3 R 9, B = R M + R R 7 R 3 R 9 + [R 6 {R 3 (R 3 R 8 + R R 9 ) - R 4 R R 9 - R 5 R 3 R 6 }] + +R R 9 (R 3 R 8 - R 4 R 7 ) + R 3 R 5 (R 5 R 7 - R 3 R 9 ), C = R M + R M [R 6 {R 3 R R 8 - R (R 4 R 8 -R 5 R 7 )}] + [R {R 3 (R 8 R 8 - R 9 R 7 ) R 7 (R 4 R 8 -R 5 R 7 ) - R R 4 R 9 + R 6 (R 4 R 9 - R 5 R 8 )}] + [R 5 {R 3 (R 8 R 4 -R 9 R )- -R 7 (R 4 R 4 - R 5 R ) + +R R 5 R 3 R 5 }], D = R M 3 + R M + R 6 R 6 (R 4 R 4 - R 5 R ) + R R (R 5 R 7 - R 4 R 8 ), E = R M 3,

23 Unqueness, recprocy heorem and varaonal prncple... M = R 7 (R 3 R 8 +R R 9 ) + R R 3 R 9 - R 8 R R 9 - R 9 R 3 R 6, 3 M = R 7 (R R 8 - R 4 R 7 ) + R (R 3 R 8 + R R 9 ) + R 6 (R 8 R 4 - R 9 R ) - R (R 8 R 8 -R 9 R 7 ), M 3 = R (R R 8 - R 4 R 7 ), A R RR RR B RRR R R R RR RR R RR RR R RR RR C R R R RR RR RRR RRR D R R 8 R R l l R ll R ll R l R ll R l l R l R l R l R l R R l l R R l R l R R R l l K Acknowledgmen One of he auhors andana Gupa s hankful o Councl of Scenfc and Indusral Research (CSIR) for he fnancal suppor. References [] J.W. Nunzao, S.C. Cown // Archve for Raonal Mechancs and Analyss 7 (979) 75. [] S.C. Cown, J.W. Nunzao // Journal of Elascy 3 (983) 5. [3] J. Sngh, S.K. Tomar // Mech. Ma. 39 (7) 93. [4] B. Sngh // Appl. Mah. Compu. 89 (7) 698. [5] H.W. Lord, Y. Shulman // Journal of Mechancs and Physcs of Solds 5 (967) 99. [6] M. Carlea, B. Sraughan // Journal of Mahemacal Analyss and Applcaons 333 (7) 4. [7] Y.Z. Povsenko // J. Therm. Sress. 8 (5) 83. [8] Y.Z. Povsenko // Journal of Mahemacal Sresses 6 (9) 96. [9] H.M. Youssef // J. Hea Transfer 3 (). [] X. Jang, M. Xu // Physca A 389 () [] Y.Z. Povsenko // Mech. Res. Commun. 37 () 436. [] M.A. Ezza // Physca B 46 () 3. [3] G. Jumare // Compu. Mah. Appl. 59 () 4. [4] M.A. Ezza // Mahemacal Modellng 35 () [5] Y.Z. Povsenko // Journal of Thermal Sresses 34 () 97. [6] R.K. Bswas, S. Sen // Journal of braon and Conrol 7 () 34. [7] D. Iesan // Aca Mechanca 6 (986) 67. [8] H.H. Sheref, H. Saleh, F. Hamza // Inernaonal Journal of Engneerng Scence 4 (4) 59. [9] M. Aouad // Journal of Thermal Sresses 3 (7) 665. [] M. Aouad // Journal of Thermal Sresses 3 (8) 7. [] H.H. Sheref, A. El-Sad, A. Abd El Laef // In. J. Sold Sruc. 47 () 69. [] A.S. El-Karmany, M.A. Ezza // Journal of Thermal Sresses 34 () 64.

24 4 Rajneesh Kumar, andana Gupa [3] C. Caeneo // C. R. Acad. Sc. 47 (958) 43. [4] G. Rusu // Bull. Polsh Acad. Sc. Tech. Sc. 35 (987) 339. [5] M. Marn // Cenc. Ma. (Havana) 6 (998) -9. [6] P. Pur, S.C. Cown // J. Elascy 5 (985) 67. [7] R.S. Dhalwal, J. Wang // In. J. Eng. Sc. 3 (994) 83. [8] M. Brsan // Lberas Mah. () 95. [9] M.D. Sharma // Aca Mechanca 9 (9) 75. [3] R. Kumar, R. Kumar // Journal of Engneerng Physcs and Thermophyscs 84 () 463. [3] K.S. Mller, B. Ross, An nroducon o he fraconal negrals and dervaves, heory and applcaons (John Wley and Sons Inc., New York, 993). [3] I. Podlubny, Fraconal dfferenal equaons (Academc press, New York, 999). [33] M.A. Bo // J. Appl. Phys. 7 (956) 4. [34] H. Kolsky, Sress waves n solds (Clarendon Press, Oxford, Dover press, New York, 963). [35] R.S. Dhalwal, A. Sngh, Dynamc Coupled Thermoelascy (Hndusan Publshng Corporaon, Inda, 98).