Thermoelastic Problem of a Long Annular Multilayered Cylinder

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1 Wold Jounal of Mechancs, 3, 3, 6- Publshed Onlne August 3 ( Theoelastc Poble of a Long Annula Multlayeed Cylnde Y Hsen Wu *, Kuo-Chang Jane Depatent of Infoaton Manageent, Oental Insttute of Technology, Tape, Chnese Tape Depatent of Appled Matheatcs, atonal Chung Hsng Unvesty, Tachung, Chnese Tape Eal: * yhwu@al.ot.edu.tw Receved May, 3; evsed June, 3; accepted June 9, 3 Copyght 3 Y Hsen Wu, Kuo-Chang Jane. Ths s an open access atcle dstbuted unde the Ceatve Coons Attbuton Lcense, whch pets unestcted use, dstbuton, and epoducton n any edu, povded the ognal wok s popely cted. ABSTRACT Theoelastc tansent esponse of ultlayeed annula cylndes of nfnte lengths subected to known nne pessue and oute sufaces coolng ae consdeed. A ethod based on the Laplace tansfoaton and fnte dffeence ethod has been developed to analyze the theoelastcty poble. Usng the Laplace tansfo wth espect to te, the geneal solutons of the govenng equatons ae obtaned n tansfo doan. The soluton s obtaned by usng the atx slaty tansfoaton and nvese Laplace tansfo. Solutons fo the tepeatue and theal stess dstbutons n a tansent state wee obtaned. It was found that the tepeatue dstbuton, the dsplaceent and the theal stesses change slghtly as te nceases. Keywods: Theoelastc; Multlayeed Annula Cylndes; Laplace Tansfoaton; Fnte Dffeence Method. Intoducton * Coespondng autho. A theal poble ases when the coposed ateals ae geneated by a sudden change n tepeatue. Shell stuctues ae wdely used n contepoay ndustes, so we ust take cae of the theal poble. The shell stuctues ay be affected due to the pessue change o the vaous tepeatue dstbutons. It s necessay to solve fo tepeatue o pessue at fst. The dynac theoelastc esponse of ccula shell apdly change of theal envonents s potant fo the desgn of any engneeng stuctues. Due to the coplexty of the govenng equatons and the atheatcal dffcultes assocated wth the soluton, seveal splfcatons have been used. Fo exaple, Sheef and Anwa [] dscussed the poble of an annula nfntely long elastc ccula. They have neglected both the neta tes and the elaxaton effects of the poble. Sheef and Anwa [] consdeed the theoelastcty poble of an nfntely long annula cylnde coposed of two dffeent ateals wth axal syety. The soluton was obtaned n the Laplace tansfo doan by usng the potental functon appoach. The pesent wok deals wth the one-densonal quasstatc coupled theoelastc pobles of an nfntely long annula ultlayeed cylnde coposed of ultlayeed dffeent ateals. The edu has a pessue at the nne laye, the tepeatue to be heated at the oute laye, wthout body foces and ntenal heat geneaton. Devatves ae appoxated by cental dffeences esultng n an algebac epesentaton of the patal dffeental equaton. By takng the Laplace tansfo wth espect to te, the geneal solutons n the tansfo doan ae fst obtaned. The fnal solutons n the eal doan can be obtaned by nvetng the Laplace tansfo.. Foulaton Ths wok deals wth the one-densonal, quas-statc coupled, theoelastc pobles of an nfntely long annula cylnde coposed of ultlayeed lanated ateals wth axal syety unde the followng assuptons: ) Mateals of each laye ae assued to be non-hoogeneous; ) Defoaton and stan satsfy the Hooke s law and sall stan theoy; 3) The coposte cylnde s constucted of ultlayeed lanates bonded togethe pefectly; 4) The edu s ntally undstubed, and wthout body foces and ntenal heat souces; 5) The edu s appled by a foce, whch s the functon of te; 6) The tepeatue at nne laye and oute laye ae the functons of te. We now consde an nfntely long annula cylnde Copyght 3 ScRes.

2 Y. H. WU, K.-C. JAE 7 ade of ultple layes of dffeent ateals. The nne and oute ad of the cylnde ae denoted by and, o espectvely. The ultlayeed coposte s assued to be heated suddenly at the nne and oute suface unde tepeatues f and f espectvely. The tansent heat conducton equaton fo the th laye n densonal fo can be wtten as (see Equaton () below) whee E and E n whch U s the adal coponent of dsplaceent, s adus, Cv and ae specfc heat and densty of ateal, and ae the Posson s ato k, k ae adal, ccufeental theal conductvty,, ae adal and ccufeental theal expanson coeffcent, E, E ae adal and ccufeental Young s odulus,, ae the tepeatue, efeence tepeatue, and s te, espectvely. If the body foces ae absent, the equaton of equlbu fo a cylnde along the adal decton can be wtten as U E U E U E E E E () The stess-dsplaceent elatons ae E U E U (3) E U E U, (4) whee, ae Lae s constant, adal and ccufeental stesses espectvely. Let the bounday condtons of ultlayeed cylnde be at at R t U, e ct t P f at R o, e ct out f t whee f, f, P, ae nne and oute suoundng tepeatues, ntal nne pessue, the ntal tepeatue at the oute laye espectvely. The non-densonal vaables ae defned as follows: T k cos k sn k cos k sn a Cv Cv ksn k cos k cos k sn b Cv Cv sn cos cos sn w Cv Cv k cos k sn t Cv R cos sn u U R Cv R f E E e E E cos sn g Cv cos sn E Cv E h E cos sn Q Cv E cos sn Q Cv Q 3 E cos sn R Cv E cos sn R Cv U k cos k sn k sn k cos Cv cos sn () U sn cos Copyght 3 ScRes.

3 8 Y. H. WU, K.-C. JAE 3R whee T, t,, u,, ae non-densonal tepeatue, te, adus, dsplaceent, adal stess and ccufeental stess fo the th laye espectvely. Substtutng the nondensonal quanttes nto the govenng Equatons ()-(4), the tansent heat conducton equaton and stess-dsplaceent elatons have the followng nondensonal fo: b u w u a T (5) t t t u e u u T T u u Q Q 3QT u u R R 3RT f g h 3. Coputatonal Pocedues (6) (7) (8) Applyng cental dffeence n Equatons (5)-(8), we ave at the followng dscetzed equatons: u u T T T T T T w u t t a b t t (9) u u u u u T T e f u g T () h whee u u u Q Q 3 u u u R R 3 QT () RT () ae de- and,,,. The Laplace tansfo of a functon fned by st s L t t e d t t Take the Laplace tansfo fo Equatons (9)-(), we obtan the followng equatons: T T T T T w a b T st ( u su ) n, n, u, n su u, n su (3) u u u u u T T T (4) e f u g h Let the suface of the cylndcal nne suface be stess fee and subect to a te-dependent tepeatue. Afte takng Laplace tansfoaton, the bounday condtons n tansfoed doan becoe P s, T fs s c at ; at out. s, T s s c and the nteface condtons ae as follows: u, s u, s whee s, s,,, q s q, s T s T, s,,,. Substtutng the bounday condtons and the nteface condtons nto Equatons (3), (4), we obtan the fol- Copyght 3 ScRes.

4 Y. H. WU, K.-C. JAE 9 lowng equaton n atx fo (see Equatons (5) below) whee 3Q a B Q a B 3Q Q w E Q Q C a F D X Y Z G E A b D 3Q a b Q k B w E 3Q a Q F 3Q w Q Q k Q k X Y Z X Q Q P Q 3 Y 3Q a b Q k 3Q a b Q k G Z G 3Q b a P T Q Q whee denotes the last laye, k the last pont, and denotes t h laye fo,3,,, (see Equaton (6) below) whee: g h H I K L M J e f e,3,, g B C T X Y Z A B C T X Y Z si s c s A B C T X Y Z A B T X Y Z E F u G D E F u G s D E F u G D E u G I J T L M u H I J K L M u T H I J T K L M u H I T K L u (5) (6) Copyght 3 ScRes.

5 Y. H. WU, K.-C. JAE Equatons (5) and (6) can be ewtten n the followng atx fos M sit su X s c (7) Y Z G s RT Qu (8) whee the atx M,, R and Q ae the coespondng atx n Equatons (5) and (6). Substtutng Equaton (7) nto (8), we have AsIT B s c (9) C DF s whee A Q R M B Q R X C Q R Y D Q R Z F Q R G Snce the atx A s a nonsngula eal atx, the atx A possesses a set of lnealy ndependent egenvectos, hence the atx A s dagonalzable. Thee exst a nonsngula tanston atx P such that P A P daga, that s, the atces A and dag A ae sla, whee the atx dag A s a dagonal atx wth eleents,,,, whee s the egenvalue of atx A. The equaton can be obtaned as P APsIP T P B s c () P C P DP F s Equaton () can be ewtten as s c s whee dag A si T B C D F () and T P T, B P B C P C, D P D F P F Fo Equaton (), the followng solutons can be obtaned edately. F B C D T s s sc s s s () By applyng the nvese Laplace tansfos to Equaton (), we get the soluton T. The egenvalue, egenvecto and nvese Laplace tansfo of atx [A] can be solved by applyng the IMSL MATH/LIBRARY suboutnes. Afte we have obtaned T, then we can use Equatons (3) and (4) to obtan the solutons T and u T PT u Q RT (3) (4) Substtutng T and u nto Equatons () and (), we obtan the adal and ccufeental stesses. 4. uecal Results and Dscussons In ths secton, we pesent soe nuecal esults of the tepeatue dstbuton n a long ultlayeed coposte hollow cylnde, and dsplaceent and theal stesses unde tepeatue changes. The nne and oute ad of the cylnde ae assued to be. and 4.5 espectvely. Fo an nfntely long annula ultlayeed cylnde, the geoety and ateal quanttes of the cylnde (n the case of thee layes, laye :E = 58E6, k =, =., =.8E 6, =.95, C v =.3 and laye :E = 3E6, k =, =.35, =.3E 6, =.53, C v =.5 and laye 3 : E = E6, k = 7, =., =.8E 6, =.9, C v =.7 ; n the case of fve layes, laye : E = 58E6, k =, =., =.8E 6, =.95, C v =.3 and laye : E = 3E6, k =, =.35, =.3E 6, =.53, C v =.5 and laye 3:E = E6, k = 7, =., =.8E 6, =.9, C v =.7 and laye 4:E = 3E6, k =, =.35, =.3E 6, =.53, C v =.5 and laye 5 : E = E6, k = 7, =., =.8E 6, =.9, C v =.7). Each laye s assued to have a dffeent thckness (n the case of thee layes, =.5, =.5 and 3 =.5; n the case of fve layes, =., =.5, 3 =., 4 =.5 and 5 =.5). The pessue of the nne suface s assued to be P =.5E6. The constant coeffcent c = c =.. The tepeatue at nne suface s assued to be 3, at oute Copyght 3 ScRes.

6 Y. H. WU, K.-C. JAE suface whch s a functon of te s assued to be to. Fgues -4 show soe nuecal esults of thee and fve layeed cylndes at te step t =.5,,, 5 and. Fgues and show the tepeatue dstbutons along adal decton fo 3 and 5 layes case. Because of the dffeence n theal conductvty and the effect of the oute laye s to be heated. As te s sall, say t =.5, the oute laye tepeatue whch s to be heated s not so oe, so the dstbuton deceasng at fst and then nceasng. Fgues 3 and 4 show the dsplaceent along the adal decton. The axu dsplaceent occued at the nteface of fst and second layes. Fgues 5 and 6 show the adal stess dstbuton along the adal decton. Fgues 7 and 8 show the ccufeental stess along the ccufeental decton. Fgue 3. Radal dsplaceent dstbuton along adal decton fo 3 layes case. Fgue. Tepeatue dstbuton along adal decton fo 3 layes case. Fgue 4. Radal dsplaceent dstbuton along adal decton fo 5 layes case. Fgue. Tepeatue dstbuton along adal decton fo 5 layes case. Fgue 5. Radal stess dstbuton along adal decton fo 3 layes case. Copyght 3 ScRes.

7 Y. H. WU, K.-C. JAE Fgue 6. Radal stess dstbuton along adal decton fo 5 layes case. Fgue 8. Ccufeental stess dstbuton along adal decton fo 5 layes case. dstbutons have been obtaned, all of whch can be used to desgn useful stuctues o achnes fo engneeng applcatons. Thee s no lt to the nube of annula layes n a cylnde. Exeplfyng nuecal esults fo thee- and fve- layeed cylndes at dffeent te steps have been pesented. The dscontnuty n ccufeental stess at each nteface was found. It was found that the tepeatue dstbuton, the dsplaceent and the theal stesses vay slghtly as the te nceases. Fgue 7. Ccufeental stess dstbuton along adal decton fo 3 layes case. A ethod based on the fnte dffeence and Laplace tansfoaton has been developed to obtan nuecal esults. The tepeatue, dsplaceent and theal stess REFERECES [] H. H. Sheef and M.. Anwa, Poble n Genealzed Theoelastcty, Jounal of Theal Stesses, Vol. 9, o., 986, pp do:.8/ [] H. H. Sheef and M.. Anwa, A Poble n Genealzed Theoelastcty fo an Infntely Long Annula Cylnde Coposed of Two Dffeent Mateals, Acta Mechanca, Vol. 8, o. -, 989, pp do:.7/bf7885 Copyght 3 ScRes.