An Inverse Transient Thermoelastic Problem Of A Thin Annular Disc

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1 Applied Mathematics E-Notes, 6(26), c ISSN Available free at mirror sites of amen/ An Inverse Transient Thermoelastic Problem Of A Thin Annular Disc Kailesh W. Khobragade, Vinod Varghese and Namdeo W. Khobragade Received 1 November 24 Abstract This paper is concerned with an inverse transient thermoelastic problem in which we need to determine the unknown temperature, displacement and stress function on the outer curved surface of a thin annular disc when the interior heat flux is known. Finite Marchi-Fasulo transform and Laplace transform techniques are used. 1 Introduction The inverse thermoelastic problem consists of the determination of the temperature of the heating medium and the heat flux of a solid when the conditions of the displacement and stresses are known at some points of the solid under consideration. This inverse problem is relevant to different industries where machinery such as the main shaft of lathe and turbine and roll of a rolling mill is subject to heating. In [1] and [2], one-dimensional transient thermoelastic problems are considered and theheatingtemperatureandtheheatflux on the surface of an isotropic infinite slab are derived. The direct problems of thermoelasticity of a thin circular plate are considered in [5, 7, 9] and inverse problems of thermoelasticity of a thin annular disc are considered in [3] and [4]. In the present problem an attempt is made to study the inverse transient thermoelastic problem to determine the unknown temperature, displacement and stress functions of the disc occupying the space D = {(x, y, z) R 3 : a (x 2 + y 2 ) 1/2 b, h z h} with known interior heat flux. Finite Marchi-Fasulo integral transform and Laplace transform techniques are used to find the solution of the problem. Numerical estimate for the temperature distribution on the outer curved surface is obtained. Mathematics Subject Classifications:74J25, 74H99, 74D99. Post Graduate Department of Mathematics, MJP Educational Campus, RTM Nagpur University, Nagpur 44 33, India. 17

2 18 An Inverse Transient Thermoelastic Problem 2 Statement Of The Problem Consider a thin annular disc of thickness 2h occupying the space D. The differential equation governing the displacement function U(r, z, t), where r =(x 2 + y 2 ) 1/2 is with 2 U r r U r =(1+υ)a tt (1) U r =atr = a and r = b, (2) where υ and a t are the Poisson s ratio and the linear coefficient of thermal expansion of the material of the disc respectively and T is the temperature of the disc satisfying the differential equation 2 T r T r r + 2 T z 2 = 1 k T t, (3) where k = K ρc is the thermal diffusibility of the material of the disc, K is the conductibility of the medium and ρ is its calorific capacity (which is assumed to be constant), subject to the initial condition the interior condition T (r, z, ) = for all a r b and h z h, (4) T (ξ,z,t) = f(z, t) r for all a ξ b, h z h and t> (5) and the boundary conditions T (a, z, t) r = u(z, t) for all h z h and t>, (6) T (b, z, t) =g(z,t) for all h z h and t>, (7) [T (r, z, t)+k 1 T z (r, z, t)] z=h = F 1 (r, t) for all a r b and t>, (8) [T (r, z, t)+k 2 T z (r, z, t)] z= h = F 2 (r, t) for all a r b and t>. (9) The functions F 1 (r, t) andf 2 (r, t) are known constants and they are set to be zero here as in other literatures [3-4, 6, 9] so as to obtain considerable mathematical simplicities. The constants k 1 and k 2 are the radiation constants on the two plane surfaces. The function f(z,t) isassumedtobeknown while the function g(z, t) isnot.

3 Khobragade et al. 19 The stress functions σ rr and σ θθ are given by σ rr = 2µ 1 r U r (1) σ θθ = 2µ 2 U r 2 (11) where µ is the Lamé elastic constant, while each of the stress functions σ rz, σ zz and σ θz are zero within the disc in the plane state of stress. The equations (1) to (11) constitute the mathematical formulation of the problem under consideration [5]. 3 Solution Of The Problem Applying the finite Marchi-Fasulo integral transform defined in [6] to the equations (3) to (7), (5) and using (8) as well as (9), we obtain d 2 T dr d T r dr a2 T n = 1 d T k dt (12) the initial condition T (r, n, ) =, (13) the boundary conditions d T (a, n, t) dr =ū(n, t), (14) T (b, n, t) =ḡ(n, t), (15) the interior condition d T (ξ,n,t) = dr f(n, t) (16) where T denotes the Marchi-Fasulo transform of T and n is the Marchi-Fasulo transform parameter, a n are the solutions of the equation [α 1 a cos(ah)+β 1 sin(ah)] [β 2 cos(ah)+α 2 a sin(ah)] = [α 2 a cos(ah) β 2 sin(ah)] [β 1 cos(ah) α 1 a sin(ah)], α 1, α 2, β 1 and β 2 are constants and f(n, t) = h h f(z,t)dz,

4 2 An Inverse Transient Thermoelastic Problem ū(n, t) = h h u(z, t)dz, =Q n cos(a n z) W n sin(a n z), Q n = a n (α 1 + α 2 )cos(a n h)+(β 1 β 2 )sin(a n h), W n =(β 1 + β 2 )cos(a n h)+(α 2 α 1 )a n sin(a n h). Applying the Laplace transform defined in [8] to the equations (12), (14) to (16) and using (13), we obtain d 2 T dr 2 the boundary conditions + 1 r d T dr q2 T =,q 2 = a 2 n + s k, (17) d T (a, n, s) dr =ū (n, s), (18) and the interior condition T (b, n, s) =ḡ (n, s), (19) d T (ξ,n,s) dr = f (n, s), (2) where T denotes the Laplace transform of T and s is a Laplace transform parameter. The equation (17) is a Bessel equation whose solution is given by T (r, n, s) =AI (qr)+bk (qr) (21) where A, B are the constants depending on n, and I (qr), K (qr) aremodified Bessel s functions of first and second kind of order zero respectively and as r tends to zero, K (qr) tends to infinity but T (r, n, s) remainsfinite. Using (18) and (2) in (21), we obtain and A = f (n, s)k (qa) ū (n, s)k (qξ) q[k (qa)i (qξ) K (qξ)i (qa)], B = f (n, s)i (qa)+ū (n, s)i (qξ) q[k (qa)i (qξ) K (qξ)i (qa)].

5 Khobragade et al. 21 Substituting these values in (21) and then inversion of Laplace transform and finite Marchi-Fasulo transform leads to T (r, z, t) = [Y(λ m a)][y(λ m ξ)] (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ (λ m a)j (λ m r) Y (λ m r)j (λ m a)] [Y (λ m ξ)] 2 (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ (λ mξ)j (λ mr) Y (λ mr)j (λ mξ)] f(n, t )e k(λ2 m +a2 n )(t t) dt ū(n, t )e k(λ2 m +a2 n )(t t) dt (22) and g(z, t) = [Y(λ m a)][y(λ m ξ)] (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ (λ m a)j (λ m b) Y (λ m b)j (λ m a)] [Y (λ m ξ)] 2 (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ [Y(λ m ξ)j(λ m b) Y(λ m b)j(λ m ξ)] f(n, t )e k(λ2 m +a2 n )(t t) dt ū(n, t )e k(λ2 m +a2 n )(t t) dt, (23) where m, n are positive integers, and λ m are the positive roots of the transcendental equations [Y (λ m a)j (λ m b) Y (λ m b)j (λ m a)] = and = h h P 2 n(z)dz = h[q 2 n + W 2 n]+ sin(a nh) 2a n [Q 2 n W 2 n] Equations (22) and (23) are the desired solutions of the given problem with β 1 = β 2 =1andα 1 = k 1, α 2 = k 2.

6 22 An Inverse Transient Thermoelastic Problem 4 Determination Of Thermoelastic Displacement Substituting the value of T (r, z, t) from the equation (22) in (1) one obtains the thermoelastic displacement function U(r, z, t): U(r, z, t) = (1 + υ)a t [Y(λ m a)][y(λ m ξ)] (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ (λ m a)j (λ m r) Y (λ m r)j (λ m a) +(1 + υ)a t (λ m ξ)j (λ m r) Y (λ m r)j (λ m ξ)] f(n, t )e k(λ2 m +a2 n )(t t) dt ] [Y (λ mξ)] 2 (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ ū(n, t )e k(λ2 m +a2 n )(t t) dt. 5 Determination Of Stress Functions Using the series expansion of T (r, z, t) in the equations (1) and (11), the stress functions are obtained as and σ rr = 2µ r (1 + υ)a t (λ m a)j (λ m r) Y (λ m r)j (λ m a) 2µ r (1 + υ)a t (λ m ξ)j (λ m r) Y (λ m r)j (λ m ξ)] σ θθ = 2µ(1 + υ)a t λ m [Y (λ ma)][y (λ mξ)] (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ f(n, t )e k(λ2 m +a2 n )(t t) dt ] λ m [Y(λ m ξ)] 2 (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ ū(n, t )e k(λ2 m +a2 n )(t t) dt (24) λ 2 m[y(λ m a)][y(λ m ξ)] (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ (λ m a)j (λ m r) Y (λ m r)j (λ m a) 2µ(1 + υ)a t (λ mξ)j (λ mr) Y (λ mr)j (λ mξ)] f(n, t )e k(λ2 m +a2 n )(t t) dt ] λ m [Y(λ m ξ)] 2 (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ ū(n, t )e k(λ2 m +a2 n )(t t) dt.(25)

7 Khobragade et al Convergence Of The Series Solution In order for the solution to be meaningful the series expressed in equation (22) should converge for all a r b and h z h, and we should further investigate the conditions which has to be imposed on the functions u(z, t), g(z, t) andf(z, t) sothat the convergence of the series expansion for T (r, z, t) is valid. The temperature equation (22) can be expressed as T (r, z, t) = M [Y(λ m a)][y(λ m ξ)] (λ 2 m + a 2 n)[[y (λ ma)] 2 [Y (λ m a)j (λ m r) Y (λ m r)j (λ m a)] M [Y (λ m ξ)] 2 (λ 2 m + a 2 n)[[y (λ m ξ)j (λ m r) Y (λ m r)j (λ m ξ)] (λ (λ ma)] 2 [Y (λ f(n, t )e k(λ2 m +a2 n )(t t) dt ū(n, t )e k(λ2 m +a2 n )(t t) dt. We impose conditions so that both T (r, n, t) andt t (r, n, t) converge in some generalized sense to g(r, n) andh(r, n) respectively as t in the Marchi-Fasulo transform domain. As per finite classic Marchi-Fasulo transform sin(a n z)andcos(a n z)are bounded, thus converges to a finite limit with β 1 = β 2 =1andα 1 = k 1, α 2 = k 2 and h selected with finite limits for our desired solution. Taking into account of the asymptotic behaviors of, positive roots λ m, and eigenvalues a n given in [6], it is observed that the series expansion for T (r, z, t) will be convergent if e k(λ2 m +a2 n )(t t ) f(n, t ) ū(n, t ) dt 1 = O λ 2 m + a 2 n k,k>. (26) Here f(n, t )andū(n, t ) in equation (26) can be chosen as one of the following functions or their combination involving addition or multiplication of constant functions, sin(ωt ), cos(ωt ), e kt or polynomials of t. Thus, T (r, z, t) isconvergenttoa limit {T (r, z, t)} r=b,z=h as convergence of a series for r = b implies convergence for all r b at any z. 7 Special Case And Numerical Result In (23), let f(z,t) =(1 e t )(z h) 2 (z + h) 2 ξ, u(z, t) =(1 e t )(z h) 2 (z + h) 2 a, α =4(k 1 + k 2 )ξ, a=1,b=2, ξ =1.5, h=1,k=.86,

8 24 An Inverse Transient Thermoelastic Problem t = 1 second, and λ m the positive roots of the transcendental equation [Y (λ m a)j (λ m b) Y (λ m b)j (λ m a)] =. Then g(z,t) α = cos 2 (a n ) cos(a n )sin(a n ) a 2 n [Y (λ m)][y (1.5λ m)][y (λ m)j (2λ m) Y (2λ m)j (λ m)] (λ 2 m + a 2 n)[[y (λ m)] 2 [Y (1.5λ m)] 2 ] [Y(1.5λ m )] 2 [Y(1.5λ m )J(2λ m ) Y (λ 2 m + a 2 n)[[y (λ m)] 2 [Y (2λ m )J(1.5λ m )] (1.5λ m)] 2 ] (1 e t )e.86(λ2 m +a2 n )(1 t) dt. (27) The (27) is calculated numerically and it is observed that as r increases the value of increases. g(z,t) α 8 Conclusion The temperature, displacements and thermal stresses on the outer curved surface of a thin annular disc have been obtained, when the interior heat flux and the other three boundary conditions are known, with the aid of finite Marchi-Fasulo transform and Laplace transform techniques. The results are obtained in terms of Bessel s function in the form of infinite Marchi-Fasulo transform series. The series solutions converge provided we take sufficient number of terms in the series. Since the thickness of annular disk is very small, the series solution given here will be definitely convergent. Any particular case can be derived by assigning suitable values to the parameters and functions in the series expressions. The temperature, displacement and thermal stresses that are obtained can be applied to the design of useful structures or machines in engineering applications. Acknowledgments. We thank the anonymous referee for useful suggestions. We are also indebted to Professor P. C. Wankhede of Nagpur University for his concrete suggestions that help to improve the presentation of our work. References [1] K. Grysa and M. J. Cialkowski, On a certain inverse problem of temperature and thermal stress fields, Acta Mechanics, 36(198), [2] K. Grysa and Z. Koalowski, One-dimensional problem of temperature and heat flux at the surfaces of a thermoelastic slab part-i, The Analytical Solutions, NUCL. Engrg 74(1982), 1 14.

9 Khobragade et al. 25 [3] N. W. Khobragade and P. C. Wankhede, An inverse steady-state thermoelastic problem of a thin annular disc, Far East J. Appl. Math., 5(3)(21), [4] N. W. Khobragade, An inverse unsteady-state thermoelastic problem of a thin annular disc in Marchi-Fasulo transform domain, Far East J. Appl. Math., 11(1)(23), [5] W. Nowacki, The state of stress in a thick circular plate due to temperature field, Bull. Sci. Acad. Polon Ser. Tech., 5(1957), 227. [6] S. R. Patel, Inverse problems of transient heat conduction with radiation, the Mathematics Eduation, 5(4)(1971), [7] S. K. Roychaudhari, A note on the quasi-static stresses in a thin circular plate due to transient temperature applied along the circumference of a circle over the upper face, Bull. Sci. Acad. Polon Ser. Tech., 2(1972), [8] I. N. Sneddon, The Use of Integral Transform, McGraw Hill Book Co., [9] P. C. Wankhede, On the quasi-static stresses in a circular plate, Indian J. Pure Appl. Math., 13(1982),