The Hyperbolic Model with a Small Parameter for. Studying the Process of Impact of a Thermoelastic. Rod against a Heated Rigid Barrier

Transcription

1 Applied Mahemaical Scieces, Vol., 6, o. 4, 37-5 HIKARI Ld, hp://dx.doi.org/.988/ams The Hyperbolic Model wih a Small Parameer for Sudyig he Process of Impac of a Thermoelasic Rod agais a Heaed Rigid Barrier Yury A. Rossikhi ad Vikor V. Shiikov Research Ceer o Dyamics of Solids ad Srucures Voroeh Sae Uiversiy of Archiecure ad Civil Egieerig -leija Okjbarja Sree 84 Voroeh 3946, Russia Federaio Copyrigh 6 Yury A. Rossikhi ad Vikor V. Shiikov. This aricle is disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. Absrac The impac of a hermoelasic rod agais a rigid heaed barrier is cosidered usig he hyperbolic heory of dyamic hermoelasiciy wih hermal relaxaio. The Laplace iegral echique wih he subseque expasio of he foud images i erms of aural fucios of he problem is uilied as a mehod of soluio. The aalyical ime-depedece of sress, emperaure ad displaceme is obaied. Keywords: Thermoelasic rod, hyperboloc heory of hermoelasiciy, relaxaio ime, fiie speed of hea propagaio Iroducio The problem of impac of a hermoelasic rod agais a rigid heaed barrier was cosidered usig differe heories of dyamic hermoelasiciy ad mehods of soluio. Thus, he exeded hermoelasiciy based o he Maxwell-Caaeo-Veroe law of hea coducio [] was uilied wihou referece o ad wih due accou for he couplig of he srai ad emperaure fields i [5] ad [3], respecively. The aalysis of he soluios cosruced i [3, 5] is ameable oly o umerical reame. Five-erm rucaed ray series soluios have bee cosruced i [3].

2 38 Yury A. Rossikhi ad Vikor V. Shiikov The Laplace iegral rasform was used as a mehod of soluio i [5]. However, he difficulies coeced wih fidig he iverse Laplace rasforms did o allow he auhor o solve he problem i a exac form. A approximae aalysis, i which "diffusive erms" were egleced, led he auhor of [5] o a coac ime formula wherei he coac ime depeds o he impac velociy ad heaig emperaure of a rigid wall. Laer, i 993, Gree ad Naghdy [4] formulaed he so-called hyperbolic heory of hermoelasiciy wihou eergy dissipaio. Thus, he idea proposed i [5] aedaed for 5 years he Gree-Naghdy heory [4], ad eabled is auhor, Yury A. Rossikhi, o solve he dyamic boudary-value problem of hermoelasiciy, resulig i he aalyical soluio for he coac sress. The problem of he impac of a hermoelasic rod agais a rigid heaed barrier wih due accou for hea exchage bewee he rod ad he wall has bee aalyically solved i [6] via he Laplace-iegral rasform mehod i combiaio wih he expasio of he desired fucios i erms of eige fucios usig he Gree-Naghdy heory of hermoelasiciy wihou eergy dissipaio. The ifluece of he couplig of he srai ad emperaure fields o he ime-depedece of he coac sress has bee aalyed. I has bee foud ha he fields couplig resuls i a icrease i he coac sress. Usig he D'Alember mehod [9], his problem has bee reaed for he cases of ucoupled ad coupled srai ad emperaure fields, respecively, i [7] ad [8] adopig oce agai he hyperbolic heory of hermoelasiciy wihou eergy dissipaio. The problem of colliear collisio of wo hermoelasic rods wih equal cross-secio ad rheological parameers bu of differe legh ad emperaure has bee cosidered i []. Thermoelasic behavior of he rods has bee described by he Gree-Naghdy heory wihou eergy dissipaio, ad D Alember soluio has bee uilied as a mehod of soluio. Sice he couplig bewee he emperaure ad srai fields could be assumed o be small, he he perurbaio echique has bee used as well. The proposed procedure has allowed he auhors o cosruc a aalyical soluio eablig o sudy he ifluece of hermoelasic parameers o he coac duraio of wo rods, as well as o obai he sress, displaceme velociy, emperaure, ad hea flow depedeces of ime ad coordiae. I he prese paper, i order o aalye he ifluece of he hermal relaxaio ime o he behavior of he coac sress durig he impac ieracio of a hermoelasic rod wih a heaed rigid wall, a small parameer, where τ is he hermal relaxaio ime, will be iroduced, resulig i he expasio of he soluio of he problem uder cosideraio i erms of his small parameer. Problem formulaio ad mehod of soluio Le us cosider a rod of he legh of l wih a hermally isulaed laeral surface movig owards a rigid barrier wih he velociy V. The emperaure of

3 Hyperbolic model wih a small parameer 39 he rigid wall is equal o T. The impac occurs a a he referece of he coordiaes. Durig impac, a oe rod s ed ( x ) here occurs hermal exchage bewee he rod ad he wall, while he oher ed ( x l) is hermally isulaed ad free from exeral forces. Assumig ha hea propagaes wih a fiie speed, we arrive a he followig se of equaios [5]: q, x c,, () q (),, x q, u (3), x,, (4) Eu,x subjeced o he iiial ad boudary codiios u, u,, ( ), (5),, u, q h( ) ( x ), q, ( x l), (6) where q is he quaiy of hea flowig hrough he rod cross-secioal area per a ime ui, c is he specific hea a cosa srai, T T is he relaive emperaure of he body, T is he iiial emperaure of he rod, E, E is he Youg modulus, is he coefficie of liear hermal expasio, is he hermal coduciviy, is he hermal relaxaio ime, is he sress, is he desiy, u is he displaceme, h is he hea rasfer coefficie from he wall o he coac boudary, T T is he differece i emperaures of he rigid wall ad impacig rod, ad a subidex afer a comma deoes he derivaive wih respec o he x coordiae or wih respec o ime. Elimiaig he value q from he eergy coservaio law () ad Fourier law () cosiderig ieria of he hea flow, as well as he value from he equaio of moio (3) ad Hooke s law (4), we fid exp, xx( ) d,, æ (7) u c u m (8), xx,, x,

4 4 Yury A. Rossikhi ad Vikor V. Shiikov where æ с, m E, ad c E / is he velociy of he elasic wave propagaio. Elimiaig exp, xx( ) d form equaio (7), as well as from equaio (7) differeiaed oe ime wih respec o, we are led o a hea rasfer equaio of he hyperbolic ype æ. (9), xx,, Applyig Laplace rasformaio o he se of equaios (8) ad (9) wih furher rasformaio from he Laplace domai o he ime domai, he soluio was preseed i [4] i he followig form: () x ( x, ) c ( )si s( ), () ( ) x ( ) x u M si Ncos l ( ) x ( ) x x si f Gsi g, E () where 4 с( ) s( ) s( )(exp exp ) d ( ), () ( ) (3), ( ) ( ), s( ) s( )[ e e ] d, lh (4) ( ) 4 M ( ) si, (5) mc N( ) s( ) k( ) d, l (6) 4 f( ) ( ) f ( )si ( ) d, (7)

5 Hyperbolic model wih a small parameer 4 ( ) 8 l g( ) G( ) ( )si ( ), g d E (8) mc f s( ) k( ) d, l (9) E g N c s ( ) ( ) ( ), () ad ( ), ( ) / /, æ, lc, k ( ) A cos B si C exp D exp, A C D, ( ) B ( C D ), C ( )( ) ( ) D. ( )( ), This soluio is valid uil he rod is i coac wih he wall, i.e. uil (, ). 3 Aalysis of he soluio for small τ - Le us aalye he soluio for he case whe he hea relaxaio process goes o raher slowly, i.e. is a large bu fiie value, or, where is a small value. The equaio (3) akes he form, ( ) i. Suppose for disicess ha 3. The he soluio of he iegral equaio (4) could be represeed i he form s( ) a, d ( 3 d) 3 s ( d) ( ) a, () where d. h

6 4 Yury A. Rossikhi ad Vikor V. Shiikov I urs ou ha here is o eed o wrie he fucio c () i he explici form, sice i is oly a par of he expressio for ( x, ) as a erm () x c ( )si ad i he expressio for g () as a erm or ha is ( ) c( ), s( ),, ( ) c ( ) 3 s( ) s( ),,,, d ( ) c( ),, () d ( d ) 3,, ( d ) ad ( ) x x x x x s ( ) s s( ). c( )si s ( ) s Subsiuig (4) i () yields (3) x x x x ( x, ) s ( ) s s ( ) s. (5) To fid he coac duraio, le us sudy he ime depedece of he coac sress E si E (, ) ( ) cos ( ) l l f d 4 g d s ( ) ( ) si ( ) ( ). (6) Uiliig feaures of periodical dela-fucios [] (see Appedix), we could fid from equaio (6)

7 Hyperbolic model wih a small parameer 43 E E (, ) f ( ) s( ) ; E E (, ) f ( ) g( ) s( ) ; E E E 3 (, ) f ( ) f ( ) g( ) s( ). l (7) Kowig he value of s (), we could fid he fucios g () ad f () Emc g( ) ; l d 4 (8) Emc l d d g( ) ; 4 Emc g( ) ( ) ; l d d 4 (3) Emc 3d 3 g( ) ( ) ; l d d d 4 (3) mc f( ) l d 4, (3) (9)

8 44 Yury A. Rossikhi ad Vikor V. Shiikov mc 3d 3d f( ) l d d d d d 3 d d, (33) mc 3d 3d f ( ) l d d d 4 3d 3d 3 3. d d (34) Cosiderig relaioships (8)-(3) ad (3)-(34), we could rewrie he coac sress i he dimesioless form ( ) (, ) ( ), ( d)( ) ( d)( ) (35) ( ) (, ) ( ), ( d)( ) ( d)( ) (36) ( )( d ) (, ) ( d ) ( ) ( d )( ) 3d d(3 ) 3 d ( d) ( ) ( ) 3 ( ), ( )( d ) (, ) d( ) ( d ) ( ) ( d )( ) d( ) 3 ( ) ( d) ( ) (37) (38)

9 Hyperbolic model wih a small parameer 45 (, ) 3 d( ) d (3 ) ( d ) ( ) ( d )( ) (3 d ) 3 ( ), ( d) ( ) (39) where (, ) (, ),,,,, E E Puig i formulas (34)-(38) yields. (, ) ( ) ( d )( ), (39) [ d(3 )] (, ) (, ) (4) ( d ) ( ) d ( ) 3 (, ) ( ), (4) ( d ) ( ) (3 d ) 3 (, ) ( ), (4) ( d ) ( ) where d, sice a equaio () should go over io he law of h hea coducio wih a fiie value of / [] q. (43),, x Formulas (39)-(4), which correspod o he case of he pure wave equaio for emperaure cq, (44), xx, coicide wih hose derived i 978 i [5]. Referece o relaioship (39) shows ha he coac sress is cosa uil he ime, bu a he mome (he ime of arrival o he coac domai of he refleced hermal wave geeraed by he acio of he icide hermal wave) i chages abruply remaiig a cosa egaive value up o he isa

10 46 Yury A. Rossikhi ad Vikor V. Shiikov (he mome of arrival a he coac oe of wo waves a a ime, amely: he refleced hermal wave geeraed by he acio of he icide elasic wave o he free rod s ed ad refleced elasic wave origiaed from he acio of icide hermal wave). A he isa he coac sress decreases abruply eiher o ero, resulig o he reboud of he rod from he barrier a, or o a fiie value, resulig i he rod s reboud a (he isa of arrival a he coac domai of he refleced elasic wave geeraed by he acio of he icide elasic wave o he free ed of he rod) [5]. The aalysis of he -depedece of he coac sress accordig o relaioships (34)-(38), which ake he dissipaio of eergy io accou, shows ha depedig o he magiude of he relaxaio ime, he reboud of he rod may occur o oly a he momes of ad, bu wihi he ime iervals ad as well. The dimesioless ime depedece of he dimesioless coac sress is preseed i Fig. for he cases whe he hermal relaxaio ime is assumed o be ifiiely large (solid lie) ad a.5 (dashed lie) a he followig magiudes of oher parameers: d.5,.7, v /. Referece o Fig. shows ha a.5 he reboud of he rod from he wall akes place a he isa lyig wihi he ime ierval, while accordig o he hyperbolic heory of hermoelasiciy wihou eergy dissipaio he process of impac ieracio will ermiae a he ime co. Variaio i he hermal relaxaio ime could resul i differe siuaios. This fac is demosraed i Table, wherei he sigs of he coac sress calculaed by formulas (34)-(38) are preseed for hiree differe combiaios of magiudes of parameers d,, v,, ad. The isa, a which he coac sress ime chages is sig from mius o plus, refers o he coac duraio co ad, hus, o he reboud of he hermoelasic rod from he heaed barrier. Referece o Table shows ha for cases ad he reboud occurs a isas wihi he ime ierval, while for cases -4, 6-8 ad he impac ieracio ermiaes a

11 Hyperbolic model wih a small parameer Figure he ime equal o co. For case 3 he coac ime is equal o co, for cases 5 ad i is equal o co, while i case 9 i lies wihi he ime ierval. Table. Numerical resuls d v (.;.5) - (-;+).5.7 (.5; ) (;5.) (5.;5.3) (5.3; ) (; ) (;.69).7 (; ) (;89.93) (86.94;87.99) (-;+).7.7 (87.99; ) (.67;.86) (.86;.89) - (-;+) 3.5 (.893;.986) Coclusio ad expadig he soluio of he Iroducig he small parameer problem uder cosideraio i erms of he small parameer, i was possible o obai he aalyical relaioships for he sress ad emperaure, ad i paricular, for he coac sress, for he iricae coac dyamic problem cocerig he impac of a hermoelasic rod agais a rigid heaed barrier. Wih a help of he small

12 48 Yury A. Rossikhi ad Vikor V. Shiikov parameer, he ifluece of he hermal relaxaio ime o he behavior of he coac sress has bee aalyed. I has bee foud ha he coac duraio of he hermoelasic rod wih he heaed rigid wall is depede o oly o he arrival o he place of coac of elasic ad hermal waves refleced from he free rod s ed, bu o he relaxaio processes occurrig i he hermoelasic rod as well. Ackowledgemes. This research was carried ou wihi he framework of he goverme ask from he Miisry of Educaio ad Sciece of he Russia Federaio, projec No. 4/9. Refereces [] D.S. Chadrasekharaiah, Hyperbolic hermoelasiciy: A review of rece lieraure, Applied Mechaics Reviews, 5 (998), hp://dx.doi.org/.5/ [] I.M. Gelfad, G.E. Shilov, Geeralied Fucios, Vol. I, Academic Press, New York, 968. [3] V.L. Gosovski, Yu.A. Rossikhi, M.V. Shiikova, The impac of a hermoelasic rod agais a rigid heaed barrier wih accou of fiie speed of hea propagaio, Joural of Thermal Sresses, 6 (993), hp://dx.doi.org/.8/ [4] A.E. Gree, P.M. Naghdy, Thermoelasiciy wihou eergy dissipaio, Joural of Elasiciy, 3 (993), hp://dx.doi.org/.7/bf44969 [5] Yu.A. Rossikhi, Impac of a hermoelasic rod agais a rigid obsacle, Sovie Applied Mechaics, 4 (978), hp://dx.doi.org/.7/bf88575 [6] Yu.A. Rossikhi, M.V. Shiikova, The impac of a hermoelasic rod agais a rigid heaed barrier, Joural of Egieerig Mahemaics, 44 (), hp://dx.doi.org/.3/a: [7] Yu.A. Rossikhi, M.V. Shiikova, D Alember s soluio i hermoelasiciy Impac of a rod agais a heaed barrier: Par I. A case of ucoupled srai ad emperaure fields, Joural of Thermal Sresses, 3 (8), hp://dx.doi.org/.8/ [8] Yu.A. Rossikhi, M.V. Shiikova, D Alember s soluio i hermoelasiciy Impac of a rod agais a heaed barrier: Par II. A case of

13 Hyperbolic model wih a small parameer 49 coupled srai ad emperaure fields, Joural of Thermal Sresses, 3 (9), hp://dx.doi.org/.8/ [9] Yu.A. Rossikhi, M.V. Shiikova, D Alember mehod i dyamic problems of hermoelasiciy, Chaper i Ecyclopedia of Thermal Sresses, Spriger Neherlads, 5, hp://dx.doi.org/.7/ _95 [] Yu.A. Rossikhi, M.V. Shiikova, V.V. Shiikov, Shock ieracio i hermoelasic rods wih emperaure srai couplig, Proceedigs of he ASME Ieraioal Mechaical Egieerig Cogress & Exposiio,, Vol. 4, Houso, Texas, USA, Pars A ad B, hp://dx.doi.org/.5/imece-8735 Appedix. Fourier series for he geeralied fucios 4 i i cos( ) ( ) ( ) ( i ), (A) i 4 i ( ) si( ) ( ) i, i (A) 4 i i ( )si( ) '( ) ( ) '( i ), (A3) i i ( ) ( )cos( ) ( ) ' i, i 4 (A4) 4 cos ( ) 3 ( ) i a i or a i i, (A5) 4 si ( ) i a i or a i i, (A6)

14 5 Yury A. Rossikhi ad Vikor V. Shiikov where () is he Dirac δ-fucio, ad i,. Received: May 7, 6; Published: Jue 5, 6