ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 Hea Concion Poblem in a Tick Cicula Plae and is Temal Sesses e o Ramp Type Heaing Isaque A Kan 1, M H Duge, Lalsing Kalsa 3 1,3 M G College, Amoi, Dis Gadcioli (MS), India ANC, Waoa, Dis Candapu (MS), India ABSTRACT: Tis pape deal wi e sudy of unseady sae emal sesses in a ick cicula plae subjeced o abiay ea supply a uppe suface of e ick cicula plae wile e lowe suface and cicula bounday of e ick cicula plae insulaed e iniially empeaue of ick cicula plae is kep a eo degee empeaue e govening ea concion equaion is solved by using inegal ansfom ecnique and e esul illusaed gapically wi e elp of special case KEYWORDS: Quasi-saic, Tansien, Temoelasic poblem, Temal Sesses, Axisymmeic Temal Sesses I INTRODUCTION Recenly Rui e al (5) [7] did emoelasic analysis of ick walled finie leng cylindes of funcionally gaded maeials and obained e esuls fo sess, sain and displacemen componens oug e ickness and along e leng ae pesened e o unifom inenal pessue and emal loading Nowacki (1957) [5] as deemined seadysae emal sesses in cicula plae subjeced o an axisymmeic empeaue disibuion on e uppe face wi eo empeaue on e lowe face and e cicula edge Roy Couday (197) (1973) [8] and Wankede (198) [11] deemined Quasi saic emal sesses in in cicula plae Gogulwa and Desmuk (5) [] deemined emal sesses in in cicula plae wi ea souces Also Tike and Desmuk (5) [1] sudied ansien emoelasic defomaion in a in cicula plae, weeas Qian & Baa (4) [6] sudied ansien emoelasic defomaion of ick funcionally ick plae unde laeal loads and obained e esuls fo adial and axial displacemens and empeaue cange moeove Sama e al (4) [9] sudied e beavio of emoelasic ick plae unde laeal loads and obained e esuls fo adial and axial displacemens and empeaue cange ave been compued numeically and illusaed gapically fo diffeen eoies of genealied emoelasiciy Also Nasse M EI- Magay (4) (5) [3] solved wo-diamensional poblems of ick plae wi ea souces in genealied emoelasiciy VS Kulkani, KC Desmuk e al [1] consideed a ick annula disc wic is subjeced o a ansien axisymmeic empeaue field on e adial and axial diecions of e cylindical coodinae sysem and deemined e expession fo empeaue, displacemen and sess funcions e o abiay ea flux on e uppe and lowe suface In is pape we conside Quasi-saic emal sesses in ick cicula plae e o amp ype eaing Sudied by Desmuk KC and Kulkani VS and discuss e emal sesses Tis pape deal wi e sudy of unseady sae emal sesses in a ick cicula plae subjeced o asymmeic abiay ea supply a uppe suface of e ick cicula plae wile e lowe suface ( / ) and cicula bounday of e ick cicula plae insulaed e iniially empeaue of ick cicula plae is kep a eo degee empeaue e govening ea concion equaion is solved by using inegal ansfom ecnique and e esul in illusaed gapically wi e elp of special case Copyig o IJIRSET DOI:11568/IJIRSET154155 1576
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 II STATEMENT OF THE PROBLEM Conside a ick cicula plae of adius a and ickness occupying space D defined by a, / / e plae is kep a eo iniial empeaue e cicula egion D1 : b a of uppe suface ( / ) subjeced o empeaue disibuion as follows T [ T(,, )] [ H( b) H( a)] fo T[ H ( b) H ( a)] fo wee H(-b) is e Heaviside funcion Te lowe suface ( / ) and e cicula edge ( a) ae emally insulaed Assume e cicula bounday of e ick cicula plae is fee fom acion unde eses moe ealisic pescibed condiion e quasi-saic emal sesses in a ick clamped cicula plae ae equied o be deemined Hea Condiion Equaion: Te empeaue of e plae a ime saisfies e Hea Condiion equaion T 1T T 1 T k (1) wi e condiion T T f ( ) fo T f () T fo a b a () T a b a (3) T a a (4) and iniial condiions T = a = (5) wee k is emal diffusiviy of e maeial of e plae and f() = [H(-b) H ( a)] Displacemen Poenial and Temal Sesses: Te diffeenial equaion and govening e displacemen poenial funcion (,, ) 1 k (6) wee k is esain coefficien and empeaue cange = T T i, T i is e iniial empeaue e displacemen poenial funcion is known as Goodie s emoelasic displacemen poenial e displacemen funcion in e cylindical coodinae sysem ae epesened by Micell s funcion M U (7) e Micell s funcion M mus saisfy wee M U 1 v M (8) M (9) 1 (1) e Componen sesses ae epesened by emoelasic displacemen poenial an Micell s funcion M as Copyig o IJIRSET DOI:11568/IJIRSET154155 1577
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 and M G k v M (11) 1 1M G k v (1) M G k v M (13) M G k 1 v M (14) wee G and ae sea molus and Poisson s aion ecepiviy Fo acion fee suface sess funcion a a e equaion (1) o (15) Consiue maemaical fomulaion of e poblem (15) III SOLUTION OF THE PROBLEM Taking e Laplace ansfomaion of e equaion (1) o (4) w and using (5) one obains, T 1 T T P T k (16) wi e condiion p T 1 e T f () a b a P (17) T a, b a (18) and T a a (19) p is Laplace ansfom paamee Now Assume T Am J m Cos m m1 () wee Jn (x) in e Bessel funcion of e fis kind of ode n and 1 ae e oos of e equaion J 1 ( n a) = (1) Subsiue equaion () in equaion (16) one obain P m m n 1 k () Now e equaion (17) in equaion () Copyig o IJIRSET DOI:11568/IJIRSET154155 1578
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 p T 1 e f ( ) A m J m Cos[ m] P m1 Muliplying e above equaion J ( m ) on bo sides and inegaion fom o a a p a T 1 e f () J md Am J mcosmd P 1 b a f ( ) H bh a a and a a J m d J ma one obain bt A m p m 1 J1 b e a m1m J ma p Cosm (3) Using equaion () and (3) in e equaion () one obain T,, p P p Cos m 1 e bt Jm J1mb k a m1 m J ma P P Cos m k (4) Hence e empeaue disibuion funcion is given by n 1Cos n 1 ( ) bt J m J1 mb T n 1 a mjma Sin n 1 m 1 k n1 m u uh u u H u e (5) Since e iniial empeaue of e plae Ti = e empeaue cange Ti T (6) Micell s funcion M Suiable fom of M saisfying (1) is given by bt k J m J1 mb M a J a m 1 m m n1 H mn Sin m Rmn m Cos m (7) H mn and R mn ae abiay consans Goodies Temoelasic Displacemen Funcion Assuming displacemen funcion (,,) wic saisfies (6) as Copyig o IJIRSET DOI:11568/IJIRSET154155 1579
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 Cos n 1 kbt Jm J1mb n 1 (,, ) a m 1 mj ma n 1 Sin n 1 n1 m 4 n1 k n ( u) 4 uh u u H u e (8) Displacemen and Temal Sesses Using e equaion (5), (6), (7) and (8) in e equaion (7), (8) and as (11) o (14) one obain e expession fo displacemen and sesses especively kbt J1m J1mb U a m1 mj ma U n1 m Cos n 1 n 1 n 1 n Sin n 1 4 uh u u H u e k n m d u 4 1 m Hmn Cos m m Rmn Cos m m sin n (9) J J kbt 1 m 1 m a m1 mj ma n1 Sin n 1 n 1 n 1 m Sin n 1 4 k n1 m u) 4 uh u u H u e m Hmn Sin m m Rmn 1 vsin m m Cos m (3) Copyig o IJIRSET DOI:11568/IJIRSET154155 158
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 m1 m m n1 1 1 m m mj m 4GkbT J b J a J a Cos n 1 n 1 n 1 Sin n 1 m 4 k n1 m u 4 u H u u H u e Cos n 1 n 1 J m Sin n 1 n1 m u u H u u H u e J 1m m Hmn m J m Cos m J 1m m Rmn mj m Cos m m Sin m v J mm Cos m (31) 1 m m1 m m n1 4GkbT J b a J a Cos n 1 n 1 J1m n 1 m Sin n 1 4 k n1 m u 4 u H u u H u e Cos n n J m Sin n 1 1 1 k n1 m u u H u u H u e Copyig o IJIRSET DOI:11568/IJIRSET154155 1581
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 1m J m Hmn Cos m m Rmn v mj mcos m Z J1m mj1m Cos m Sin m (3) 4GkbT J m J1 mb a m1 mj ma n1 3 3 Cos n 1 ( n 1 (n 1) m Sin n 1 4 k n1 m u u H u u H u e Cos n 1 n 1 Sin n 1 k n1 m u u H u u H u e 3 3 m Hmn Cos m m Rmn 1 vcos m m Sin m (33) 4GkbT J1 m J1 mb a m1 mj ma n1 m Sin n 1 n 1 n 1 m Sin n 1 4 n1 km u 4 u H u u H u e 3 m Hmn Sin m 3 mrmn 4 vsin m m Cos m (34) Copyig o IJIRSET DOI:11568/IJIRSET154155 158
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 Deeminaion of Unknown Abiay funcion H mn and R mn In ode o saisfy e condiion (15) and solving e equaion (31) and (34) one obain 1 Hmn 3 mcos m Cos n 1 n 1 Cosn 1 n 1 Sin n 1 Sin n 1 m 4 Rmn k n1 m 4 u H u u H u e (35) (36) IV NUMERICAL RESULTS, DISCUSSION AND REMARKS Te numeical calculaion ave been caied ou fo seel (SN 5C) plae wi paamees a = 1m, b=a/=5, =1m, =/Temal diffusiviy k=159x1-6(m s-1) wi 1 = 38317, = 7156, 3 = 11735, 4 = 13337, 5 = 1647, 6 = 196159, 7 = 761, 8 = 5937, 9 = 9468, 1 = 318 ae e oos of anscendenal equaion J 1 ( n a) = In ode o examine e influence of amp-ype eaing on e uppe suface on ick cicula plae, one pefomed e numeical = -5,, 5, 5 Numeical vaiaions in adial diecions ae sown in e figues wi e elp of compue pogamme V CONCLUSION In is cape, a ick cicula plae is consideed and deemined e expessions fo empeaue, displacemen and sess funcion e o amp-ype eaing on e uppe suface as a special case maemaical model is consuced fo f ( ) [ H( 5) H( 1)] and pefomed numeical calculaions Te emoelasic beavio is examined suc as empeaue, displacemen and sesses In figue 1 : Fo = Tempeaue is gaally is decease and inceases and en suddenly dop ill = 7 and again incease ill = 8 afe an mainain same saus ill 1 Fo = 5 Tempeaue is negaive and gaally inceases fom = o = 65 afe a i again incease and as a value posiive fom = 65 o 1 Fo = 5 Tempeaue is negaive fom = o = 65 wi damping naue Afe a i will become posiive damping saus ill = 1 In figue : Goodie poenial funcion gaally inceases fom = o = 1 Fo = 5 o 5 Te displacemen funcion is e gaally deceases inceases in adius of e ick cicula plae In figue 3 : Fo = e adial displacemen U deceases fom = o = 5 and en i incease ill = 1 Fo = 5 o = 5 Te adial displacemen U gaally incease fom = o = 5 and e deceases o = 1 Te adial displacemen U as maximum value a = 5 In figue 4 : Fo = e axial displacemen U is consan ougou e cicula plae Te axial displacemen U is inceases fom = o = 3 In figue 5 : Fo = e sess funcion is deceases fom = o = 1 Fo = 5 o = 5 e sess funcion is inceases fom = o = 1 In figue 6 : Fo = e sess funcion gaally inceases fom = o = 1 Copyig o IJIRSET DOI:11568/IJIRSET154155 1583
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 5E+15 4E+15 T 3E+15 E+15 1E+15 E+ = -1E+15 = - 4 6 8 1 -E+155, 5-3E+15= -4E+155 In Figue 1 Tempeaue T vesus fo diffeen values of a = - 5, 5 (Seel SN5C) Tempeaue of e cicula plae vaies wi e adius and ickness of e cicula plae 8E+8 6E+8 4E+8 E+8 E+ -E+8-4E+8-6E+8-8E+8-1E+9 = 5= -5, 5 = 4 6 8 1 In Figue Displacemen funcion vesus fo diffeen values of a = - 5, 5 (Seel SN5C) Te displacemen funcion as e maximum displacemen a e op of e cicula plae I is vay wi e ickness and adius of e cicula plae 3E+9 E+9 U 1E+9 E+ -1E+9 -E+9-3E+9-4E+9 = 5 = -5, 5 4 6 8 1 = In Figue 3 Radial Displacemen U vesus fo diffeen values of a = - 5, 5 (Seel SN5C) Te adial displacemen is diecly vaies wi e ickness and adius of e cicula plae Copyig o IJIRSET DOI:11568/IJIRSET154155 1584
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 15E+4 U 1E+4 5E+3 E+ = -5E+3 4 6 8 1-1E+4-15E+4 -E+4 = - -5E+4 = 5 5 In Figue 4 Axial Displacemen U vesus fo diffeen values of a = - 5, 5 (Seel SN5C) Te axial displacemen is vay wi e ickness and adius of cicula plae and as consan value a = as well as =1 = 5 3E+ E+ 1E+ = E+ -1E+ = - 4 6 8 1 5, 5 -E+ =5 In Figue 5 Sess Funcion vesus fo diffeen values of a = - 5, 5 (Seel SN5C) (=, 4 1) Te sess funcion is decease a eo wi e ickness weeas a =1 i will incease wi ickness of e cicula plae 3E+ E+ 1E+ = E+ -1E+ = - 4 6 8 1 5, 5 -E+ = 5 In Figue 6 Sess Funcion vesus fo diffeen values of a = - 5, 5 (Seel SN5C) Te sess funcion deceases a e cene of e cicula plae wee as i will incease wi adius and ickness of e cicula plae Copyig o IJIRSET DOI:11568/IJIRSET154155 1585
ISSN(Online): 319-8753 ISSN (Pin): 347-671 Inenaional Jounal of Innovaive Reseac in Science, Engineeing and Tecnology (An ISO 397: 7 Ceified Oganiaion) Vol 4, Issue 1, Decembe 15 REFERENCES [1] A K Tike, K C Desmuk, Tansien emoelasic defomaion in a in cicula plae J Adv Ma Sci Appl 15 (1) (5) [] Gogulwa V S and Desmuk K C, Temal sesses in a in cicula plae wi ea souces, Jounal of Indian Academy of Maemaics, Vol 7, No 1, 5 [3] Nasse M, El-Magaby, Two dimensional poblem wi ea souces in genealied emoelasiciy wi ea souce, Jounal of Temal sesses, Vol 7, pp 7-39, 4 [4] N Noda, F Asida, T Tusuji, An invese ansien emoelasic poblem of a ansvesely isoopic body J Applied Mec (Tans ASME Se E) 56 (4) (1998) 791-797 [5] Nowacki W, Te sae of sesses in a ick cicula plae e o empeaue field, Bull Acad Polon, Sci, Sc Scl Tec, Vol5, pp 7, 1957 [6] Qian L F, and Baa R C, Tansien emoelasic defomaion of a ick funcionally gaded plae J Tem Sesses, 7(4), pp 75-74 [7] Rui M, Angosaai A and Nagdabadi R, Temoelasic analysis of ick walled finie leng cylindes of funcionally gaded maeial, Jounal of Temal Sesses, Vol 8, pp 391-48, 5 [8] Roy Couday S K, A noe of quasi saic sess in a in cicula plae e o ansien empeaue applied along e cicumfeence of a cicle ove e uppe face, Bull Aca Polon Sci, Se, Scl, Tec, -1,197 [9] Sama J N Sama P K and Sama R L, Beavio of emoelasic ick plae unde laeal loads, Jounal of emal sesses, Vol 7, pp 171-191, 4 [1] V S Kulkani, K C Desmuk Sadana, Vol 3, Pa5, pp 561-575 (7) [11] Wankede PC, On e Quasi saic emal sesses in a cicula plae, Indian Jounal of Pue and Applied Maemaics, Vol 13(11), pp 173-177, Nov 198 Copyig o IJIRSET DOI:11568/IJIRSET154155 1586