Transient Thermoelastic Behavior of Semi-infinite Cylinder by Using Marchi-Zgrablich and Fourier Transform Technique

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1 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ Transin Thrmolasic Bhavior of Smi-infini Cylindr by Using Marchi-Zgrablich and Fourir Transform Tchniqu Badrinah E. Ghong 1,1, Kiriwan P. Ghadl 1, 1 Dparmn of Mahmaics, Dr. B. A. Marahwada Univrsiy, Aurangabad , (M.S.) India. {badrighong, drkp.ghadl}@gmail.com Absrac. Th prsn work dals wih ransin hrmolasic problm of a smi infini hollow cylindr o drmin h mpraur, displacmn and hrmal srsss wih h sad condiions. Th ransin ha conducion quaion wih sad condiions is solvd by using Marchi-Zgrablich ransform and Fourir Sin ransform simulanously and h rsuls for mpraur disribuion, ha flux disribuion, displacmn and hrmal srss funcions ar obaind in rms of infini sris of Bssl's funcion. Ths rsuls solvd for spcial cas by using Mah-Cad 007 sofwar and prsnd graphically by using Origin sofwar. Kywords: Smi-infini hollow cylindr, Transin ha conducion, Thrmal srsss, Marchi-Zgrablich and Fourir Sin ransform. 1 Inroducion Th Thrmolasic problms ar on of h mos frqunly ncounrd problms by sciniss. Th wid variis of problms ha ar covrd undr conducion also mak i on of h mos rsarchd and hough abou problms in h fild of nginring and chnology. This kind of problms can b solvd by various mhods. Ths problms consis of drminaion of unknown mpraur, ha flux, displacmn and hrmal srss funcions of solids whn h condiions of mpraur and displacmn and srss ar known a h som poins of h solid undr considraion. Sirakowski and Sun [1] sudid h dirc problms of fini lngh hollow cylindr and drmind an xac soluion. Grysa and Cialkowski [] and Grysa and Kozlowski [3] discussd on dimnsional ransin hrmolasic problms drivd h haing mpraur and h ha flux on h surfac of an isoropic infini slab. Furhr Dshmukh and Wankhd [4] sudid an axisymmric invrs sady sa problm of hrmolasic dformaion o drmin h mpraur, displacmn and srss funcions on h our curvd surfac of fini lngh hollow cylindr. Rcnly 30

2 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ Wald and Khobragad [5] obaind h soluion for ransin hrmolasic problm of fini lngh hollow cylindr. In his work w considr smi-infini hollow cylindr. Th ransin ha conducion quaion is solvd by using fini Marchi-Zgrablich ingral ransform as in Marchi and Zgrablich [6] and Fourir Sin ransform as dfind in Snddon [7] simulanously and h rsuls for mpraur disribuion, unknown ha flux, displacmn and hrmal srsss ar obaind in rms of infini sris of Bssl's funcion and i is solvd for spcial cas by using Mahcad-007 sofwar and illusrad graphically by using Origin sofwar. Th Mahmaical Modl Considr a hollow cylindr occupying spac D as dfin and as shown in figur 1 blow. D : ( x, y, z) a r x y b,0 z Th hrmolasic displacmn funcion is govrnd by h Poisson s quaion as in [5] (1 ) (1 ) at (1) wih 0 a r a and r b () whr 1, and a r r r z ar h Poisson s raio and linar cofficin of hrmal xpansion of h marial of h cylindr and T is h mpraur of h cylindr saisfying h diffrnial quaion as in Ozisik [8] T 1T T 1 T r r r z k (3) subjc o h iniial condiion T( r, z, ) 0 0 (4) and h boundary condiions T ( r, z, ) T( r, z, ) 0 r ra (5) T ( r, z, ) T( r, z, ) 0 r rb (6) T ( r, z, ) T( r, z, ) 0 z z0 (7) 31

3 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ T ( r, z, ) T( r, z, ) 0 z (8) z whr k is h hrmal diffusiviy of h marial of h cylindr. Th radial and axial displacmns U and W saisfying h uncoupld hrmolasic quaions ar U 1 (1 ) (1 ) U a T (9) r r (1 ) r 1 (1 ) T W (1 ) a (10) z (1 ) z U U W whr is h volum dilaion and r r z U (11) r W (1) z Th srss funcions ar givn by rz ( a, z, ) 0, rz ( b, z, ) 0, rz ( r, z,0) 0, (13) and r ( a, z, ) p1, r ( b, z, ) p0, z ( r,0, ) 0, (14) whr p 1 and p0 ar h surfac prssurs assumd o b uniform ovr h boundaris of h cylindr. Th boundary condiions for h srss funcions (13) and (14) ar xprssd in rms of h displacmn componns by h following rlaions: U U W r ( G) r r z (15) z ( G) W U U z r r ( ) U G U W r r z (16) W U rz G r z (18) G whr is h Lam s consan, G is h shar modulus and U and W ar 1 h displacmn componns. Th quaions (1) o (18) consiu h mahmaical formulaion of h problm undr considraion. (17) 3

4 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ Figur 1: Gomry of Smi-infini Cylindr. 3 Th Soluion 3.1 Drminaion Tmpraur Funcion T( r, z, ) Applying fini Marchi-Zgrablich ransform [6] o h quaion (3), on obains d T 1 dt m T (19) dz k d Furhr applying Fourir Sin ransform [7] o h quaion (19), on obains * dt * kp T 0 (0) d p n m whr Equaion (0) is h firs ordr linar diffrnial quaion, whos soluion is givn by * T C1. kp (1) whr C 1 is consan. Applying invrsion of Fourir Sin ransform [7] and Marchi-Zgrablich ransform [6] o h quaion (1), w obain C (,, ) 1S sin 0 k1 k mr kp T nz () m 1 m whr S p( k1, k, mx) J p( mx) G p( k1, ma) Gp( k, mb) Gp ( mx ) J p ( k1, ma ) J p ( k, mb ) 33

5 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ 3. Drminaion of Thrmolasic Displacmn Subsiuing h valu of T( r, z, ) from quaion () in quaion (1), on obains h hrmolasic Displacmn ( r, z, ) funcion as (1 ) C1S 0( k1, k, ) (,, ) sin mr r kp r z a nz (3) (1 ) m1 4m Using quaion (3) in quaion (11) and (1), on obains h radial and axial displacmn U and W as (1 ) a C1S 0( k1, k, r) r sin m kp U nz (1 ) 4 m1 m r ' ms0 ( k1, k, mr ) rs0 ( k1, k, mr ) (4) (1 ) a C1S 0( k1, k, r) r cos m kp W n nz (5) (1 ) 4 m1 m 3.3 Drminaion of Srss Funcions Using quaions (4) and (5) in quaions (15) o (18), h srss funcions ar obaind as (1 ) a 3 ' C sin 1 kp r G nz 4 r ms0( k1, k, mr) S0( k1, k, mr) (1 ) 4 m1 m r 4 (,, ) '' (,, ) 4 ms0 k1 k mr S0 k1 k mr r ms0 ( k1, k, mr) 6 r (,, ) (,, ) 4 3 (,, ) ' ms0 k1 k mr S0 k1 k mr r ms0 k1 k mr S0( k1, k, mr) (1 ) a (,, ) C sin 1rS0 k1 k mr kp n (1 ) 4 m 1 z m ' r ms0 ( k1, k, mr ) rs0 ( k1, k, mr ) (1 ) a C (,, ) sin 1S0 k1 k mr r kp n n (1 ) 4 m 1 z m (6) (1 ) a C1S 0( k1, k, mr) r kp z G n sin n z (1 ) 4 m1 m (1 ) a C1 kp 3 ' sin nz 4 r ms0( k1, k, mr) S0( k1, k, mr) (1 ) 4 m1 m 34

6 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ 3 ' m 0 1 m 0 1 m m 0 1 m 0 1 m 6 r S ( k, k, r) S ( k, k, r) 4 r S ( k, k, r) S ( k, k, r) (1 ) a C1rS 0( k1, k, mr) sin nz (1 ) 4 m1 m r ' ms0 ( k1, k, mr ) rs0 ( k1, k, mr ) kp (7) (1 ) a C1rS 0( k1, k, mr) kp G sin n z (1 ) 4 m1 m ' r ms 0 ( k 1, k, mr) rs 0 ( k 1, k, mr) (1 ) a C1 kp 3 ' sin nz 4 r ms0( k1, k, mr) S0( k1, k, mr) (1 ) 4 m1 m 4 '' 4 m 0 1 m 0 1 m m 0 1 m r S ( k, k, r) S ( k, k, r) r S ( k, k, r) 3 ' m 0 1 m 0 1 m m 0 1 m 0 1 m 6 r S ( k, k, r) S ( k, k, r) 4 r S ( k, k, r) S ( k, k, r) 1 0( 1,, m ) kp n sin nz (8) m1 m (1 ) a C S k k r r (1 ) 4 (,, ) ' (,, ) (1 ) a C rs k k r S k k r r rz G n cosn z (1 ) 4 m1 m (1 ) a C1S 0( k1, k, mr) r ncos nz (1 ) 4 m1 ' r ms 0 ( k 1, k, mr) rs 0 ( k 1, k, mr) m m 0 1 m kp m kp (9) 4 Spcial Cas and Numrical Rsuls For h spcial cas smi infini cylindr is mad up of sl marial wih innr radius a 1uni and our b unis. A numrical rsuls for quaions (), (3), (6)-(9) ar obaind by mahmaical sofwar Mahcad and ar dpicd graphically as shown in following figur

7 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ Figur. Tmpraur disribuion Figur5. Normal srss disribuion Figur3. Displacmn disribuion Figur6. Tangnial srss disribuion Figur4. Radial srss disribuion Figur7. rz disribuion 5 Conclusion In his papr, h mpraur disribuion, displacmn and srss funcions hav bn invsigad wih h hlp of ingral ransform chniqus. Th xprssions ar rprsnd graphically. Figur show h mpraur variaion in h cylindr along radial dircions. Figur 3 show h displacmn dvlops in h cylindr along radial dircions. Figur 4-7 show h srsss occur in h smi infini cylindr along h radial dircions. 36

8 Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : Volum 1 Issu 5 (May 01) hp:// hps://sis.googl.com/si/ijmsjournal/ Th rsuls ha ar obaind can b applid o h dsign of usful srucurs and can b us in sudy of oil ranspor. Rfrncs 1. Sirakowski, R. L. and Sun, C. T.: An Exac Soluion o h Elasic Dformaion of A Fini Lngh Hollow Cylindr, Journal of h Franklin Insiu, vol. 86, pp (1968). Grysa, K. and Cialkowski, M. J.: On A Crain Invrs Problm of Tmpraur and Thrmal Srss Filds, Aca Mchanica, vol. 36, pp (1980) 3. Grysa, K. and Kozlowski, Z.: On Dimnsional Problm of Tmpraur and Ha Flux Drminaion a h Surfacs of A Thrmolasic Slab Par-I, Th Analyical Soluions, Nuclar Enginring and Dsign, vol. 74(1), pp (1983) 4. Wald, R. T. and Khobragad, N. W.: Transin Thrmolasic Problm of A Fini Lngh Hollow Cylindr, Canadian Journal of Scinc and Enginring Mahmaics, vol. 3(), pp (01) 5. Dshmukh, K. C. and Wankhd, P. C.: An Axisymmric Invrs Sady-Sa Problm of Thrmolasic Dformaion of A Fini Lngh Hollow Cylindr, Far Eas. J. Appl. Mah, vol. 1(3), pp (1997) 6. Marchi, E. and Zgrablich, G.: Ha Conducion in Hollow Cylindr wih Radiaion, Proc. Edingburgh Mah. Soc., vol. 14(), pp (1964) 7. Snddon, I. N.: Th Us of Ingral Transforms, Mcgraw-Hill Company, Nw York (197) 8. Ozisik, N. M.: Boundary Valu Problm of Ha Conducion, Inrnaional Txbook Company, Scranon, Pnnsylvania (1968) 37