An inverse steady state thermal stresses in a thin clamped circular plate with internal heat generation

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1 Americn Journl of Engineering Reserch (AJER) e-issn : p-issn : Volume-02, Issue-10, pp Reserch Pper Open Access An inverse stedy stte therml stresses in thin clmped circulr plte with internl het genertion C. M. Bhongde 1 nd M. H. Durge 2 1 Deprtment of Mthemtics, Shri. Shivji College, Rjur, Mhrshtr, Indi 2 Deprtment of Mthemtics, Annd Niketn College, Wror, Mhrshtr, Indi Abstrct: - This pper dels with the determintion of temperture, displcement nd therml stresses in thin clmped circulr plte with internl het genertion. A clmped circulr plte is subjected to rbitrry known interior temperture. Under stedy stte, the fixed circulr edge nd lower surfce of circulr plte re thermlly insulted. Here we modify Kulkrni (2008) nd designed most generl solution of displcement potentil, rdil stresses nd ngulr stresses. The governing het conduction eqution hs been solved by the method of integrl trnsform technique. The results re obtined in series form in terms of Bessel s functions. The results for temperture chnge, displcement nd stresses hve been computed numericlly nd illustrted grphiclly. Keywords: - Inverse problem, thin clmped circulr plte nd internl het genertion. I. INTRODUCTION During the second hlf of the twentieth century, nonisotherml problems of the theory of elsticity becme incresingly importnt. This is due to their wide ppliction in diverse fields. The high velocities of modern ircrft give rise to erodynmic heting, which produces intense therml stresses tht reduce the strength of the ircrft structure. The inverse thermoelstic problem consists of determintion of the temperture of the heting medium, the het flux on the boundry surfces of the thin clmped circulr plte when the conditions of the displcement nd stresses re known t the some points of the thin clmped circulr plte under considertion. Nod et l. (1989) discussed n nlyticl method for n inverse problem of three dimensionl trnsient thermoelsticity in trnsversely isotropic solid by integrl trnsform technique with newly designed potentil function nd illustrted prcticl pplicbility of the method in engineering problem. Sbherwl K. C. (1965) studied n inverse problem of het conduction. Greys et l. (1989) investigted n inverse temperture field problem of theory of therml stresses. Deshmukh nd Wnkhede (1998) studied n inverse trnsient problem of qusi sttic therml deflection of thin clmped circulr plte. Ashid et l. (2002) studied the inverse trnsient thermoelsticity problem for composite circulr disc constructed of trnsversely isotropic lyer. Most recently Bhongde nd Durge (2013) considered thick circulr plte nd discuss, effect of Michell function on stedy stte behvior of thick circulr plte, now here we consider thin clmped circulr plte with internl het genertion subjected to rbitrry known interior temperture. Under stedy stte, the fixed circulr edge nd lower surfce of circulr plte re thermlly insulted. Here we modify Kulkrni (2008) nd designed most generl solution of displcement potentil, rdil stresses nd ngulr stresses. The governing het conduction eqution hs been solved by the method of integrl trnsform technique. The results re obtined in series form in terms of Bessel s functions. A mthemticl model hs been constructed for thin clmped circulr plte with the help of numericl illustrtion by considering luminum (pure) circulr plte. No one previously studied such type of problem. This is new contribution to the field. The inverse problem is very importnt in view of its relevnce to vrious industril mechnics subjected to heting such s the min shft of lthe, turbines nd the role of rolling mill. II. FORMULATION OF THE PROBLEM Consider thin clmped circulr plte circulr plte of thickness 2h defined by 0 r, z. Let the plte be subjected to rbitrry known interior temperture f(r) within region < z <. With lower w w w. j e r. o r g Pge 276

2 surfce z = nd circulr surfce r = re thermlly insulted. Under these more relistic prescribed conditions, the unknown temperture g r, which is t upper surfce of the plte. Temperture, displcement nd stresses in thin clmped circulr plte with internl het genertion re required to be determined. Following Roy Choudhuri (1972), ssume tht for smll thickness h plte is in plne stte of stress. In fct the smller the thickness of the plte compred to its dimeter, the nerer to plne stte of stress is the ctul stte. Then the displcement equtions of thermoelsticity hve the form U i,kk + 1+v 1+v e 1 v,i = 2 1 v tt,i (1) e = U k,k ; k, i = 1,2. (2) where U i - displcement component e - dilttion T temperture nd v nd t re respectively, the Poisson s rtio nd the liner coefficient of therml expnsion of the plte mteril. Introducing U i = U,i i = 1, 2. we hve 2 1 U = 1 + v t T (3) 1 2 = k 1 k 2 σ ij = 2μ U,ij δ ij U,kk, i, j, k = 1, 2. (4) where μ is the Lmes constnt nd δ ij is the Kronecker symbol. In the xilly-symmetric cse U = U r, z, T = T r, z nd the differentil eqution governing the displcement potentil function U r, z is given s 2 U + 1 r 2 r U = 1 + v r t T (5) = 0, r = for ll time t. (6) U = U r The stress function σ rr nd σ θθ re given by σ rr = 2μ U (7) r r σ θθ = 2μ 2 U (8) r 2 In the plne stte of stress within the plte σ rz = σ zz = σ θz = 0 (9) Temperture T(r, z) of the circulr plte stisfying het conduction eqution s follows, 2 T + 1 T + 2 T + q = 0 (10) r 2 r r z 2 k with the conditions T r T z = 0 t r =, z (11) = 0 t z =, 0 r (12) T = f r known t z = ξ, ξ, 0 r (13) nd T = g r unknown t z =, 0 r (14) where k is the therml conductivity of the mteril of the plte, q is the internl het genertion. Equtions (1) to (14) constitute the mthemticl formultion of the problem. III. SOLUTION To obtin the expression for temperture T ( r, z ), we introduce the finite Hnkel trnsform over the vrible r nd its inverse trnsform defined s in Ozisik (1968) T β m, z = r K 0 β m, r T(r, z) dr (15) r =0 T(r, z) = m=1 K 0 β m, r T β m, z (16) w w w. j e r. o r g Pge 277

3 where K 0 β m, r = 2 J 0 (β m r ) J 0 (β m ) nd β 1, β 2.. re roots of the trnscendentl eqution J 1 β m = 0 (17) where J n x is Bessel function of the first kind of order n. On pplying the finite Hnkel trnsform defined in the Eq. (15) nd its inverse trnsform defined in Eq. (16) to the Eq. (10), one obtins the expression for temperture s T r, z = m =1 2 J 0 β m r J 0 β m A β m, ξ F β m cosh β m z+h + A β m, z where A β m, z is prticulr integrl of differentil Eq. (10) nd F β m is Hnkel trnsform of f r. The unknown temperture g r cn be obtined by substituting z = in Eq. (18) s g r = m =1 2 J 0 β m r J 0 β m A β m, ξ F β m 1 da β m, cos 2β m cos β m +ξ sin β m ξ + A β cos β m +ξ m, To obtin displcement potentil U r, z using Eq. (18) nd Eq. (5), one obtin, U r, z = 2 J 1 + v 0 β m r J 0 β m t m=1 2 β m J0 β m A β m, ξ F β m cosh β m z+h + A β m, z Now using Eqs. (18) nd (20) in Eq. (7) nd (8), one obtins the expressions for stresses respectively s σ rr = 2 2μ J 1 + v 0 β m r r t m =1 β m J 0 β m σ θθ = 2 2μ A β m, ξ F β m cosh β m z+h 1 + v t + A β m, z J 1 β m r m =1 J 0 β m A β m, ξ F β m cosh β m z+h + A β m, z (18) (19) (20) (21) (22) IV. SPECIAL CASE AND NUMERICAL CALCULATIONS 1. f r = r 2 (23) pplying finite Hnkel trnsform s defined in eq.(15) to the eq.(23), one obtins F β m = 2 J 0 β m J 1 β m 2J 2 β m 2. q r, z = δ r r 0 δ z z 0 q β m, z = 2 r 0 δ z z J 0 β m r 0 0 J 0 β m where δ r is well known dirc delt function of rgument r. = 1m, = 0.1m, r 0 = 0.5m nd z 0 = 0.05m. 4.1 Mteril Properties The numericl clcultion hs been crried out for luminum (pure) plte with the mteril properties defined s Therml diffusivity α = m 2 s 1, Specific het c ρ = 896 J/kg, Therml conductivity k = W/m K, Sher modulus G = 25.5 G p, Poisson rtio θ = 0.281, 4.2 Roots of trnscendentl eqution The β 1 = , β 2 = , β 3 = , β 4 = , β 5 = , β 6 = re the roots of trnscendentl eqution J 1 β m = 0. The numericl clcultion nd the grph hs been crried out with the help of mthemticl softwre Mt lb. w w w. j e r. o r g Pge 278

4 V. DISCUSSION In this problem, thin clmped circulr plte is considered which is subjected to rbitrry known interior temperture nd determined the expressions for unknown temperture, displcement nd stresses. As specil cse mthemticl model is constructed for f r = r 2 nd performed numericl clcultions. The thermoelstic behvior is exmined such s temperture, displcement nd stresses in thin clmped circulr plte with internl het genertion. Fig. 1 Temperture T(r, z) in rdil direction Fig. 2 Displcement U(r, z) in rdil direction w w w. j e r. o r g Pge 279

5 Fig. 3 Rdil stress σ rr in rdil direction Fig. 4 Angulr stress σ θθ in rdil direction From Fig. 1 Due to internl het genertion temperture is incresing for 0 r 0.2, 0.4 r 0.6 nd decresing for 0.2 r 0.4, 0.6 r 0.8 long rdil direction. The overll behvior of temperture is decresing nd it is inversely vry with rbitrry known interior temperture long rdil direction. From Fig. 2 Due to internl het genertion displcement is decresing for 0 r 0.2, 0.4 r 0.6 nd incresing for 0.2 r 0.4, 0.6 r 0.8 long rdil direction. The overll behvior of displcement is decresing nd it is inversely vry with rbitrry known interior temperture long rdil direction. From Fig. 3 Due to internl het genertion the rdil stress is decresing for 0.2 r 0.6 nd incresing for 0.6 r 0.8 long rdil direction. The overll behvior of rdil stress is compressive nd it is inversely vry with rbitrry known interior temperture long rdil direction. From Fig. 4 Due to internl het genertion the ngulr stress is incresing for 0.2 r 0.4, 0.6 r 0.8 nd decresing for 0.4 r 0.6 long rdil direction. The overll behvior of ngulr stress is tensile nd it is inversely vry with rbitrry known interior temperture long rdil direction. w w w. j e r. o r g Pge 280

6 VI. CONCLUSION Due to internl het genertion temperture nd displcement re decresing nd it is inversely vry with rbitrry known interior temperture long rdil direction. Due to internl het genertion the rdil stresses re compressive nd the ngulr stresses re tensile nd It is inversely vry with rbitrry known interior temperture long rdil direction. The results obtined here re useful in engineering problems prticulrly in the determintion of stte of stress in thin clmped circulr plte, bse of furnce of boiler of therml power plnt nd gs power plnt. REFERENCES [1] Nod N., Ashid F. nd Tsuji T, An inverse trnsient thermoelstic problem of trnseversly isotropic body, Journl of pplied Mechnics, 56(4), 1989, [2] Sbherwl K. C., An inverse problem of trnsient het conduction, Indin Journl of Pure Applied Physics, 3E(10), 1965, [3] Grys K., Cilkowski M. J. nd Kminski H., An inverse temperture field problem of theory of therml stresses, Nucl. Eng. Des..64, [4] Deshmukh K.C. nd Wnkhede P. C., An inverse trnsient problem of qusi-sttic therml deflection of thin clmped circulr plte, Bulletin of pure nd pplied sciences, 17E(1), [5] Ashid F. nd Skt S., Tuchert, T. R. nd Ymshit Y., Inverse trnsient thermoelstic problem for composite circulr disc, Journl of therml stresses, 25, 2002, [6] Bhongde C. M. nd Durge M. H., Effect of Michell function on stedy stte behvior of thick circulr plte, IOSR J. of Mthemtics, 8(2), 2013, [7] Kulkrni V. S., Study of some thermoelstis problem, Thesis, R. T. M. Ngpur University, Ngpur, [8] Roy Choudhry S. K., A note of qusi sttic stress in thin circulr plte due to trnsient temperture pplied long the circumference of circle over the upper fce, Bull Acd Polon Sci, Ser, Scl,Tech, 20, 1972, 21. [9] M. N. Ozisik, Boundry Vlue Problems of Het Conduction, Interntionl Text Book Compny, Scrnton, Pennsylvni, w w w. j e r. o r g Pge 281