Quasi-Static transient Thermal Stresses in a Robin's thin Rectangular plate with internal moving heat source

Transcription

1 Weely Scence Research Journal Orgnal Artcle Vol-1, Issue-44, May 014 Quas-Statc transent Theral Stresses n a Robn's n Rectangular late w nternal ovng heat source D. T. Solane and M.. Durge ABSTRACT Keywords: Robn's n rectangular late, ovng heat source, eral stresses, Green's eore. D. T. Solane and M.. Durge Fro Ths aer concern w e transent non-hoogeneous eroelastc roble Sudhaar Na and Uashanar Khetan w Robn s boundary condton n n rectangular late of sotroc ateral, college Aola, Maharashtra State, Inda. Anand Netan Collge, Warora, occuyng regon R: 0 where a, 0 y b, 0 z c b c a, c b, s very sall as Maharashtra State, Inda. coared to b. Intal teerature of e late s f, y, z and late s laced n an abent teerature zero. The late s subjected to e actvty of ovng ont Artcle Is Publshed On May 014 Issue & Avalable At heat source at ont ovng, y, z w constant velocty vector u u1 u j u3 where u1, u, u 3 are e coonent of velocty vector along, y, z aes resectvely. The heat conducton equaton contanng heat generaton ter s solved by DOI : / /10013/53 alyng ntegral transfor technque and Green s eore s adoted n dervng e soluton of heat conducton equaton. The soluton s obtaned n a seres for of trgonoetrc functon and e eral stresses are derved..introducton: Durng e second half of 0 century, non-soeral robles of e eory of elastcty becae ncreasngly ortant. Ths s due to er wde alcaton n dverse felds. The hgh veloctes of odern arcraft gve rse to aerodynac heatng, whch roduces ntense eral stresses at reduce e streng of arcraft structure. Recently D. T. solane, S. M. Durge have studed e eral stresses, n n sold cylnder and hollow cylnder w Drchlet's, Neuann's and Robn's boundary condton and rectangular lates fro [1] to [6]. Nobody revously have studed such tye of ree densonal teerature dstrbuton and eral stresses w ovng heat source n n rectangular late w Robn's boundary condton. Ths s new contrbuton to e feld of eroelastcty. In s resent aer we deterne teerature dstrbuton and eral stresses n n rectangular late w ovng ont heat source w Robn's tye boundary condton. The heat conducton equaton contanng heat generaton ter s solved by alyng ntegral transfor technque and Green's eore s adoted n dervng e soluton of heat conducton equaton. The soluton s obtaned n a seres for of trgonoetrc functon and e eral stresses are derved. 3.Forulaton of e roble: Consder a n rectangular late of sotroc ateral of leng a, bread b and heght c ; occuyng e regon R: 0 where a,0 y b,0 z c b a, c b, c s very sall as coared to b. Intal teerature of e late s f, y, z laced n an abent teerature zero. The late s subjected to e actvty of nstantaneous ovng ont heat source at e ont whch, y, z changes ts lace along, y, z aes ovng w constant velocty vector u where u1 u j u3 u 1, u, u 3 are Page No-1

2 Vol-1, Issue-44, May 014 Quas-Statc transent Theral Stresses n a Robn's n Rectangular late w nternal ovng heat source coonent of velocty vector along, y, z aes resectvely. The actvty of ovng heat source or ntal teerature of e late ay cause e generaton of heat due to nuclear nteracton at ay be a functon of oston and te n e for g, y, z, t w/s 3. The teerature dstrbuton of e rectangular late s descrbed by e dfferental equaton of heat conducton w heat generaton ter as n [7] age no. 9 s gven by 1 1 T T g t where T T, y, z s teerature dstrbuton, s eral conductvty of e ateral of e late, s eral dffusvty, s densty, c s secfc heat of e ateral c ence above equaton reduces to W ntal and boundary condton and g s voluetrc energy heat generaton ter. s Lalacan oerator n rectangular coordnates n ree denson. Now consder an nstantaneous ovng ont heat source at a ont and, y, z releasng ts heat sontaneously at te. Such voluetrc ovng heat source n rectangular coordnates s gven by g, y, z, t g y y z z t 1 1 T T g y y z z t 3.1 t Where, u1t y u t, z u3t 3. T T T y T ht 0 y T z T ht 0 z at at a 3.4 at y at y a 3.6 at z at z a 3.8 T f, y, z at t 0, Forulaton of e eroelastc roble: Let us ntroduce a eral stress functon related to coonent of stress n e rectangular coordnates syste as n [8] where, c cs coleentary soluton and s artcular soluton. cand are governed by a lnear hoogeneous dfferental equaton and lnear non-hoogeneous dfferental equaton 4 0 c 4.1 E 4. Page No-

3 Vol-1, Issue-44, May 014 Quas-Statc transent Theral Stresses n a Robn's n Rectangular late w nternal ovng heat source Where s teerature change T T where T s ntal teerature y snce late s n z s neglgble y y y 4.5 The boundary condton s at 0 y b4.6 5.Soluton: We defne ntegral transfor of And ts nverse ntegral transfor by y T, y, z, t by T,,, t T, y, z, t X Y y Z z dv 5.1 R T, y, z, t T,,, t X Y y Z z 0 5. Where X c o s sn 5.3 Y y cos y sn y 5.4 Z z c o s z sn z 5.5 h 5.6 s roduct of e noralzaton ntegral and a b c N N N 5.7 are roots of e transcendental equaton are roots of e transcendental equaton tan a 5.8 tan b 5.9 are roots of e transcendental equaton tan c 5.10 Tang ntegral transfor of equaton 3.1 and usng boundary condton and followng Green's eore we obtan T N T dv T dv T ds R R 1 s n n 5.11 Page No-3

4 Vol-1, Issue-44, May 014 Quas-Statc transent Theral Stresses n a Robn's n Rectangular late w nternal ovng heat source d T dt T g X Y y Z z t t T f,, g X Y y Z z e e Tang nverse ntegral transfor we obtan T X Y y Z z t e Where f,, g X Y y Z z e 5.13 X Y y Z z f t Where f t e t Soluton of Theroelastc roble: Let e sutable for of Let e sutable for of c satsfyng 4.1 be y y y y c 0 y Ae Be cos y Ce De sn 6.1 satsfyng 4. be X Y y Z z E f t 6. y y y y X Y y Z z 0 y Ae Be cos y Ce De sn E f t 6.3 Fro 4.3 and 6.3 we obtan y A e y B y e A e B e y cos y C e y D e C e y D e sn 0 E X Y y Z z f t 6.4 Fro 4.4and 6.3 we obtan y y y y E X Y y Z z 0 y Ae Be cos y Ce De sn f t 6.5 Fro 4.5 and 6.3 we obtan y y sn y cos y Ae B e A e B e y Ce D e C e D e y 0 E Y y Z z f t [ sn c o s ] Alyng condton 4.6 to 6.5 and 6.6 we obtan 6.6 Page No-4

5 Vol-1, Issue-44, May 014 Quas-Statc transent Theral Stresses n a Robn's n Rectangular late w nternal ovng heat source E Y b Z z e A f t 1 b b b b b E Y b Z z e B f t 1 b b b E Y b Z z e C f t 1 b b b E Y b Z z e D f t 1 b Substtutng ese value n above equatons we obtan E Y b Z z f t 0 b snh cosh snh cosh snh cosh cos y y b b y b b y b y b b y b b yb y snh y b b cosh y b b snh y b cosh y b b snh y b b cosh yb sn E X Y y Z z f t 6.11 b co sh y b snh y b b snh y b y cos Y b EZ z y y f t b b cosh y b snh y b b snh y b y sn 0 X Y y 6.1 E Y b Z z y f t b ycosh y b b snh y b bcosh y b snh y b b cosh y b b snh y b sn snh cosh cosh cosh snh snh cos y b y b y b b y b b y b b y b y b 8. References: E X Y y Z z f t D. T. Solane, M.. Durge Quas-Statc transent Theral Stresses n a Drchlet's n Sold cylnder w nternal ovng heat source IOSR-JM Volue 10,Issue ver.1 Mar-Ar.014 PP D. T. Solane, M.. Durge Quas-Statc transent Theral Stresses n a Neuann's n Sold cylnder w nternal ovng heat source AJER Volue-03, Issue-03, D. T. Solane, M.. Durge Quas-Statc transent Theral Stresses n a Robn's n Sold cylnder w nternal ovng heat source AJCEM 3 : March-Arl D. T. Solane, M.. Durge Quas-Statc Transent Theral Stresses n a Drchlet's n ollow Cylnder w Internal Movng eat Source Internatoal journal of Physcs and Maeatcs ISSN: vol.4 1 January- March D. T. Solane, M.. Durge Quas-Statc Transent Theral Stresses n a Neuann's n ollow Cylnder w Internal Movng eat Source 6.D. T. Solane, M.. Durge Quas-Statc Transent Theral Stresses n a Robn's n ollow Cylnder w Internal Movng eat Source. Internatonal journal of Maeatcs Trends and Technology vol. 8 nuber 1- Arl M. Necat Ozs `eat conducton, Second Edton, A Wley-Interscence Publcaton John Wley and Sons, nc. New-Yor. 8. N. Noda, R. B. etnars, Y. Tangawa, Theral Stresses, second edton, 00 Page No-5