Effect of Heat Generation on Quasi- Static Thermal Stresses in a Solid Sphere

Transcription

1 IOS Jounl of Mthetics (IOS-JM) e-issn: ,p-ISSN: X, Volue 7, Issue 5 (Jul. - Aug. 3), PP -9 Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee S.P. Pw, K.C. Deshukh, G.D. Ked 3 S. N. Mo nd St. G. D. Sf Science College Tus Dist- Bhnd (M. S. ), Indi. Deptent of Mthetics,.T.M. Ngpu Univesity, (MS.) Indi. Deptent of Mthetics, Kl Nehu Mhvidhyly, Ngpu, (M. S.) Indi. Abstct: In this ppe, genel nlysis of one diensionl non-stedy stte tepetue distibution nd stesses unde thel lod in solid sphee subjected to diffeent types of het souces is developed. The ticle dels with coptive study of the effect of vying het genetion on displceent nd thel stesses. The het conduction eqution solved by integl tnsfos technique with convective thel boundy condition nd bity initil nd suounding tepetue. The esults e obtined in tigonoetic seies nd e studied nueiclly nd e illustted gphiclly. Keywods: Thel stesses, het souces, integl tnsfo. I. Intoduction The study of the pobles of deteintion of thel stesses in diffeent solids unde the diffeent thel condition nd het souces becoe subject of extensive esech ll ound the globe. Het conduction in spheicl objects is n ipotnt poble in engineeing pctices. Tnsient thel stesses in sphee e discussed by nube of uthos. Although the theoelsticity hs been well undestood fo oe thn centuy, ely studies e focused on the theoeticl ppoch. The histoy of litetue on these pobles found in the texts by Pkus [], Boley, Wiene [], Nowcki [3], Nod [], Cslw nd Jege [5]. Duing the lst two decdes incesed ttention hs been given to tnsient pobles, especilly to those involving cylindicl nd spheicl geoeties with het genetion. The genetion of het hs significnt effect on the tepetue pofile nd its effect on thel stesses in solids in the Engineeing fields nd life sciences. Cslw nd Jege [5] studied the use of souces nd sinks in cses of vible tepetue in sphee, Ozisik [6] discussed ny hoogeneous nd non hoogeneous het conduction boundy vlue pobles with het souces, Cheung et l [7] studied the tnsient poble in sphee with locl heting, Tkeuti et l [8] studied the tnsient thel stesses of hollow sphee due to otting het souce, Hetnski [9] discussed the stesses in long cylinde due to otting line souce, Nsse M. EI-Mghby [,] del with pobles of theoelsticity with het souces. Deshukh et l [] studied the deteintion of displceent nd thel stesses in thin hollow cicul disk due to intenl het genetion nd integl tnsfo is used to solve the het distibution nd stesses e obtined in Bessel s functions. Deshukh et l [3] studied the thel deflection which is built in-edge in thin hollow disk subjected to the ctivity of het souce which chnges its plce on the plte sufce with tie. ecently Ked nd Deshukh [] deteined thel stesses in thin clped hollow disk unde unstedy tepetue field due to point het souce. In this ppe, the one diensionl qusi-sttic uncoupled theoelstic poble of solid sphee with het genetion is consideed. The i is to obtin the theticl odel fo pedicting the esults bout the tepetue pofile nd stesses with consideing independently diffeent types of het souces within body nd ssuing bity initil nd suounding tepetues. The specil cses e studied with instntneous point, volue nd spheicl het souces. This is new ppoch to hve knowledge of coptive study of het distibution nd poduced stesses in sphee due to intenl het souces. Integl tnsfo technique is to obtin tepetue distibution. This is novel ppoch of study of thel stesses which is useful in engineeing field whee diffeent types of souces e to be used. II. Foultion Of The Poble Conside the solid sphee defined by. Initilly the sphee is kept t bity tepetuef(). Fo tie t > het geneted within the sphee t the te of g, t J/s 3 nd het is dissipted by convection fo the boundy t = to the ediu t tepetuef(t). The sphee is hoogeneous nd isotopic. The tepetue distibution, displceent nd thel stesses e to be deteined nd nlyse gphiclly. The tnsient tepetue distibution is govened by [6] the following eqution, T + T g (,t) + = T in <, t > () k α t Pge

2 Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee Boundy nd initil condition k T + T = f(t) t =, t > () T = F in t t = (3) A new dependent vible U(, t) is defined s U, t = T(, t) () Then the poble (-3) is tnsfoed s U g (,t) + = U k α t in <, t > (5) U = t = in (6) U f (t) + MU =, t =, t > (7) k U = F() in, t = (8) whee, H =, M = H (9) k whee, k is thel conductivity, is het tnsfe coefficient nd α is thel diffusivity of the teil. The tepetue is syetic with espect to cente of sphee, function of tht is the dil distnce only. One diensionl poble of theoelsticity ens spheiclly syetic poble [], in which the sheing stesses nd stins vnish nd stin nd stess coponents in spheicl coodinte θ nd diection e identicl ς θθ = ς, θθ = () ς θ = ς θ = ς = () θ = θ = = () The equilibiu eqution without body foce in spheicl coodintes [] is educes to ς + ς ς θθ ς φφ = dς d + ( ς ς θθ ) = (3) Stess stin eltion o Hooke s eltions e ς = µ + λe βτ () ς θθ = ς φφ = µ θθ + λe - βτ (5) whee, stin diltion e = + θθ + = + θθ (6) ς, ς θθ nd ς φφ e the stesses in the dil nd tngentil diection nd nd θθ e stins in dil nd tngentil diection, τ is the tepetue chnge, e is the stin diltion nd λ nd µ e the Lé constnts elted to the odulus of elsticity E nd the Poisson s tio ν s, υe λ =, µ = E υ υ υ (7) The stin coponent in tes of dil displceent u = u is = du d θθ = = u (8) The boundy condition on tction fee sufce is ς = t = (9) Now with equtions (-9) one cn obtin the displceent nd thel stesses s [] α υ τ τ d 3 3 τ 3 [(+ υ) ς = αe υ [ d+ (-υ) τ 3 τ d] () d] () τ d τ] () ς θθ = ς = αe [ d+ υ 3 The equtions (-) constitutes the Mtheticl foultion of the poble III. Anlytic Solutions Following the genel pocedue of Ozisik [6], we develop the Fouie integl tnsfo of U(, t) ove the vible in poble (-8) nd the invese foul s (Integl Tnsfo) U (, ) = K( ) U(, t)d (3) = (Invese Foul) U, t = = K(, ) U(, ) () Whee, the sution is tken ove ll Eigen vlues. On pplying the bove integl tnsfo nd invese foul to the poble (5-8), one obtins, the expession fo the tepetue function of non-hoogeneous boundy poble of het conduction in solid sphee s, Pge

3 T, t = e = t sin( ) Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee +M +M +M = F( )sin + t e α t t = k = g, t sin d ksinβf(t )dt (5) K (, ) = sin N + M + M = + M + M + M sinβ (6) N = β + M M = H, finite vlue H = (7) k e the positive oots of the tnscendentl eqution cot = M The oots of this tnscendentl eqution e el if (H ) > H > (8) The tepetue chnge is obtined s τ = T, t F = e = t sin( ) +M +M +M = F( )sin + t e t t = k = g, t sin d ksinβf(t )dt F() (9) Using Equtions (-), Displceent nd stesses e obtined s, υ υ ( = e t sin cos N L F() d) + υ 3=e βt sinβ βcosββnl F()d (3) + + ς = E υ 3 3 ς θθ = ς φφ = E υ whee, L = = e t sin cos = β N e t sin cos = β N 3 e t sin cos = β N 3 e t sin cos = β N e t sin = N F( )sin + e t t t k L F() d L F() d) L F() d + L F() d) L F() g, t sin d = + sin β k f(t ) dt (3) (3) (33) IV. esults nd Discussion The exct nlyticl solutions fo tepetue, displceent nd thel stesses e obtined in the pevious pt. The theticl softwe MATLAB is used fo futhe nueicl clcultion nd gphicl nlysis. Fo specil cses we ssue the initil tepetue F =, theefoe τ = T(, t) nd fo siplicity tke the bient tepetuef t = t. The nueicl solutions e pesented fo following teil popeties, Thel diffusivity α =.3 6 s Thel conductivity k = 386W/k Specific het c ρ = 383 J/kgK Poisson s tio υ =.35 Setting the dius of the sphee = nd M =., the oots of the tnscendentl eqution cot = M e s [6] β =.7593, β =.5379, β 3 = 7.75, β =.95, β 5 =.8, β 6 = 7.3 Then tepetue chnge eq. (9) educes to, 3 Pge

4 Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee τ, t = = e t sin( ) t + e t k t = + M + M + M = = g, t sin d Following cses e independently discussed with diffeent types of souces, F( )sin + k sin f(t ) dt (3) Cse Let het souce is instntneous constnt volue souce of stength g i J/ tht eleses its het spontneously t t = i.e. single explosion tkes plce within the sphee nd the enegy elesed thoughout the solid sphee. It is elted with volue het souce by the eltion s [6] g, t = g i δ(t ), theefoe using (3, 3, 3 nd 3) τ(, t) = k α υ k +M e = t sin( ) = +M +M ς = α E υ k +M = +M +M ς θθ = ς φφ = α E υ k υ sin cos g i +M sin cos sin cos g i +M +M sin cos = g i sin cos +M +M +M g i sin cos 3 + sin ( t )e t + α + υ sin cos (35) + sinβ ( t )e t + α β e t (36) sin ( ) cos ( 3 sin cos + sinβ ( t )e t + α β e t (37) + sin ( ) cos ( 3 sin ( + sinβ ( t )e t + α β e t (38) Cse The het souce is instntneous point het souce of stength g pti (J) situted t the cente of the sphee nd eleses its het spontnteniously t tie t =. this souce is elted with voluetic het souce [6] by g, t = g pti π δ t δ (39) T(, t) = α υ k k ς = α E υ k = e = t sin( ) = ς θθ = ς φφ = α E υ k υ sin cos +M +M +M g pti π sin cos +M +M +M = +M +M +M g pti π sin cos g pti π +M +M +M g pti π + υ sin cos + sin t e t + () + sin ( t )e t + α e t () sin ( ) cos ( 3 + sin ( t )e t + α e t () + sin ( ) cos ( 3 sin ( + sin ( t )e t + α e t (3) Cse3 The het souce is instntneous spheicl souce of dius of totl stength g spi J situted concenticlly inside the sphee nd eleses its het spontneously t tie t = τ then s [6] g, t = g spi δ t τ δ π T(, t) = e t β sin(β k = ) +M g spi e β β +M τ sin β +M π + sinβ( βt )e βt + β () Pge

5 Tep Tep Tepetue α υ k ς = α E υ k = +M = +M +M ς θθ = ς φφ = α E υ k +M +M +M Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee υ sin cos +M +M +M sin cos + υ sin cos g spi e τ sin β π + sinβ ( t )e t + β e t (5) sin ( ) cos ( 3 g spi e τ sin β π + sinβ ( t )e t + α β e t (6) = sin cos + sin ( ) cos ( 3 sin ( g spi e τ sin β π + sinβ ( t )e t + α β e t (7) 3 t=. t=.5 t= Figue : Tepetue cse t=. t=.5 t= Figue: Tepetue cse 5 5 g=5 =.8 tu=5 t=. t=.5 t= Figue 3: Tepetue cse Pge

6 Displceent Displceent Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee Fig, nd 3 epesent the tepetue vition fo the instntneous constnt volue, instntneous point nd instntneous spheicl souce espectively. In evey cse hs the souce of constnt gnitude g = 5 but the ntue is diffeent. The tepetue vition long the dil diection is shown in the gphs. In cse the tepetue deceses long the dil diection nd thee e soe dii whee it hs the constnt vlue nd independent of tie. Tepetue is lowest ne the cente of the sphee nd tie psses fist it ieditely inceses nd hs locl iniu nd xiu peks. Fo the cuve ssocited with tie t =. the tepetue deceses long the dil diection. In cse the tepetue is highest bout the cente nd it gee with the instntneous point souce t the cente of the sphee. The tepetue deceses long the dil diection. It is obseved tht fo vey sll tie it chnges the diection lso. Fo the spheicl souce eployed t =.8 nd t = τ =, the tepetue deceses long the dil diection. It hs the xiu vlues t bout the dius =. nd iniu vlue t the cente of the sphee. The tepetue distibution chnges with the chnge in the ntue of the souce. 8 6 t=. t=.5 t= Figue : Displceent Cse t=. t=.5 t= Figue 5: Displceent cse Fig, 5 nd 6 shows the chnge in the displceent long the dius. In cse it is seen tht the displceent inceses long the dil diection. Fo vey sll tie the incese is line. As tie psses thee is vition in the tepetue distibution nd xiu displceent occu ne the sufce of the sphee nd the displceent t the cente is vey ino.. Fo the instntneous point souce t the cente, fig 5 shows the gete displceent bout the cente nd highest vlues occu t bout the dius =.5, while fo the spheicl souce the xiu vlues of the shifts towds the sufce. The displceent is ino t the cente of the sphee in cse nd 3.In cse 3 the displceent is independent of the tie t bout =.. Fo cse nd 3 the displceent is lge on the sufce but fo cse conditions is evesed nd it gee with the point het souce t the cente. 6 Pge

7 dil Stess dil Stess dil Stess Displceent Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee 35 g=5 tu= = t=. t=.5 t= Figue 6: Displceent cse 3 9 g= t=. t=.5 t= Figue 7: dil stess cse 5 dil Stess fo =.3* - 6 M=-. g=5 v= t=. t=.5 t= Figue 8: dil stess cse t=. t=.5 t= Figue 9: dil Stess cse Pge

8 Hoop Stess Hoop Stess Hoop Stess Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee 5 3 t=. t=.5 t= Figue : Hoop stess cse t=. t=.5 t= Figue : Hoop stess cse t=. t=.5 t= Figue : Hoop stess cse 3 Fig 7, 8 nd 9 shows the vition of dil stesses long the dil diection fo thee diffeent types of het genetion within the sphee espectively. Fo the gph it is vey uch cle tht the stesses on the sufce e null s pe the ssued echnicl condition induced fo the dil stesses on the sufce of sphee. Fo instntneous volue het souce within the sphee the stesses e copessive inside the sphee nd the vlues deceses gdully on the sufce. The tension is lge nd s good s constnt fo <.5 nd the ntue of the vition is ino. Fo cse the ntue of the stesses is copessive s well s tensile. Thee is lge copession on the sufce fo cse, while thee is tension on the sufce fo cse Pge

9 Effect of Het Genetion on Qusi- Sttic Thel Stesses in Solid Sphee Fig, nd shows the vition of tngentil stesses long the dil diection. Fo instntneous voluetic souce the stesses e copessive fo vey sll tie. As tie psses the ntue of the stesses buptly chnges, cente of the sphee is unde tension fo cse, nd thee e cetin dii whee the stess is independent of tie while the sufce hs got copession. The ntue of the stesses continuously chnges fo tensile to copession nd copession to tensile nd stess vlues e highest bout the cente nd gdully deceses on the sufce. Fo the point het souce, the cente is unde copession nd stesses inceses long the dil diection with vition. Like cse, the thee e cetin dii whee the stesses e constnt. Fo the spheicl souce the vition of tngentil stess is shown in fig. V. Conclusions In this study the nlyticl solutions e obtined fo tepetue distibution, displceent nd stesses unde thel lod with bity initil nd bient tepetue nd nlysis is de by eploying thee diffeent het souce cses nd esults e obtined independently. In the nlysis instntneous point, volue nd spheicl souce e used to ke the coptive study. In obsevtions it is found tht thee is totl chnge in the tepetue nd thel stesses pofile long the dius with chnge in the ntue of the souces. This odel cn be pplied to spheicl stuctues nd to design useful stuctul pplictions. The poposed ethod y be edily extended to solve wide nge of physicl engineeing pobles with chnge in the fo of bity initil nd suounding tepetue. The nueicl esults e discussed s specil cses. Acknowledgeent The uthos e thnkful to Univesity Gnts Coission, New Delhi to povide the ptil finncil ssistnce unde jo/ino esech poject schee. efeences [] Pkus H., Instionäe Wäespnnungen, Spnge,Wien, (959). [] Boley B.A. nd Weine J.H., Theoy of thel stesses, Wiley, New Yok, (96). [3] Nowcki W., The stte of stess in thick cicul plte due to tepetue field, Bull Sci. Acd. Polon Sci. Tech., 5, 7, (957). [] Nod N., Hetnski.B., Tnigw Y., Thel Stesses, nd Ed. Tlo nd Fncis, New Yok, 3, (3). [5] Cslw H.S. nd Jege J.C., Conduction of het in solids, Clendon Pess, nd Ed (959). [6] Ozisik M.N., Boundy Vlue Poble Of Het Conduction, Intentionl text book Copny, Scnton, Pennsylvni (968). [7] Cheung J.B., Chen T. S. nd Thiuli. K, Tnsient thel stesses in sphee by Locl heting, J. Appl. Mech., (), (97). [8] Tkiuti Y. nd Tnigw Y., Tnsient thel stesses of hollow sphee due to otting het Souce, J. The. Stesses, 5(3-), (98). [9] Hetnski.B., Stesses in long cylinde due to otting line souce of het, AIAA, J., 7 (3), 9-3, (969). [] Nsse M., EI-Mghby, Two diensionl poble in genelized theoelsticity with het souces, J. of Thel Stesses, 7, 7-39, (). [] Nsse M., EI-Mghby, Two diensionl poble fo thick plte with het souces In genelized theoelsticity, J. of Thel Stesses, 8, 7-, (5). [] Kulkni V.S., Deshukh K.C., Wbhe S.D., Qusi-Sttic Thel Stesses Due to Het Genetion in Thin Hollow Cicul Disk, J. of Thel stesses, 3(8), , (8). [3] Deshukh K.C., Khndit M.V. nd Kulkni V.S., Thel stesses due to tepetue Distibution in hollow disk heted by oving het souce, F Est J. of Applied Mthetics, 66(), 5-37, (). [] Ked G.D. nd Deshukh K.C., Deteintion of thel stesses in thin clped hollow disk unde unstedy tepetue field due to point het souce, IOS J. of Mthetics, (6), -9, (3). 9 Pge