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An inverse quasi-static steady-state thermal stresses in a thick circular plate
ARTICLE IN PRESS
Journal of the Franklin Institute 345 (2008) 29–38 www.elsevier.com/locate/jfranklin
An inverse quasi-static steady-state thermal stresses in a thick circular plate V.S. Kulkarnia, K.C. Deshmukhb, a
Department of Mathematics, Government College of Engineering, Chandrapur 442 401, Maharashtra, India b Post-Graduate Department of Mathematics, Nagpur University, Nagpur 440 010, Maharashtra, India Received 6 November 2006; received in revised form 1 June 2007; accepted 11 June 2007
Abstract The present paper deals with the determination of unknown temperature and thermal stresses on the upper surface of a thick circular plate. A thick circular plate is subjected to arbitrary known interior temperature under steady state. The fixed circular edge and lower surface of the circular plate are thermally insulated. The governing heat conduction equation has been solved by using Hankel transform methods. The results are obtained in series form in terms of Bessel’s functions. The results for displacement and stresses have been computed numerically and illustrated graphically. r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Steady state; Thermoelastic problem; Inverse; Thermal stresses
1. Introduction During the second half of the twentieth century, nonisothermal problems of the theory of elasticity became increasingly important. This is due to their wide application in diverse fields. The high velocities of modern aircraft give rise to aerodynamic heating, which produces intense thermal stresses that reduce the strength of the aircraft structure. Noda et al. [1] discussed an analytical method for an inverse problem of threedimensional transient thermoelasticity in a transversely isotropic solid by integral transform technique with newly designed potential function and illustrated practical applicability of the method in engineering problem. Ashida et al. [2] studied the inverse Corresponding author.
E-mail addresses: [email protected] (V.S. Kulkarni), [email protected] (K.C. Deshmukh). 0016-0032/$32.00 r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2007.06.003
ARTICLE IN PRESS 30
V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
transient thermoelastic problem for a composite circular disc constructed of transversely isotropic layer. Grysa and Kozlowski [3] investigated an inverse one-dimensional transient thermoelastic problem and obtained the temperature and heat flux on the surface of an isotropic infinite slab. Sabherwal [4] studied an inverse problem in heat conduction. Deshmukh and Wankhede [5] solved the inverse thermoelastic problem in an anisotropic cylinder and determined the temperature and thermal stresses on the outer curved surface of a cylinder. Also, Deshmukh and Wankhede [6–8] studied inverse problem on hollow cylinders, circular plates and annular disc by applying integral transform technique and illustrated them numerically. Wankhede [9] determined quasi-static thermal stresses in thin circular plate. Recently Gogulwar and Deshmukh [10] solved the inverse problem of thermal stresses in a thin annular disc. This paper deals with the realistic problem of inverse quasi-static steady-state thermal stresses in a thick circular plate, which is subjected to arbitrary interior temperature. The fixed circular edge and lower surface of the circular plate are thermally insulated. The unknown temperature and thermal stresses on the upper surface of thick circular plate is required to be determined. Due to unknown temperature on the upper surface of the thick circular plate, it expands in axial direction and bends concavely at the center. This expansion is inversely proportional to thickness of circular plate, which is a new and novel contribution of this paper. The results presented here are more useful in engineering problem particularly in the determination of the state of strain in circular plate constituting foundations of containers for hot gases or liquids, in the foundations for furnaces, etc. 2. Formulation of the problem Consider a circular plate of thickness 2h occupying space D defined by 0prpa, hpzph. Let the plate be subjected to arbitrary known interior temperature f(r) within region hozoh, with lower surface z ¼ h and circular surface r ¼ a thermally insulated. Under these more realistic prescribed conditions, the unknown temperature g(r) which is at the upper surface of plate z ¼ h and quasi-static thermal stresses due to unknown temperature g(r) are required to be determined. The differential equation governing the displacement potential function f(r, z) is given in [11] as q2 f 1 qf q2 f þ þ ¼ Kt, qr2 r qr qz2
(1)
where K is the restraint coefficient and temperature change t ¼ T T i , Ti is the initial temperature. Displacement function f is known as Goodier’s thermoelastic displacement potential. The steady-state temperature of the disc satisfies the heat conduction equation q2 T 1 qT q2 T þ 2 ¼ 0, þ qr2 r qr qz
(2)
with the conditions qT ¼0 qr
at r ¼ a;
hpzph,
(3)
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
qT ¼ 0 at qz
z ¼ h;
0prpa,
T ¼ f ðrÞ ðknownÞ at z ¼ x;
31
(4)
hpxoh; 0prpa,
(5)
and T ¼ gðrÞ
ðunknownÞ at z ¼ h;
0prpa.
(6)
The displacement functions in the cylindrical coordinate system are represented by the Michell’s function defined in [11] as ur ¼
qf q2 M , qr qr dz
(7)
uz ¼
qf q2 M þ 2ð1 nÞr2 M 2 . qz qz
(8)
The Michell’s function M must satisfy r2 r2 M ¼ 0,
(9)
where r2 ¼
q2 1 q q2 þ þ 2. 2 r qr qz qr
(10)
The component of the stresses are represented by the thermoelastic displacement potential f and Michell’s function M as 2 q f q q2 M 2 srr ¼ 2G vr Kt þ M , (11) qr2 qz qr2
1 qf q 1 qM Kt þ vr2 M , r qr qz r qr
(12)
2 q f q q2 M 2 ð2 vÞr Kt þ M qz2 qz qz2
(13)
syy ¼ 2G
szz ¼ 2G and
q2 f q q2 M 2 þ ð1 vÞr M 2 srz ¼ 2G , qrqz qr qz
(14)
where G and n are the shear modulus and Poisson’s ratio, respectively. For traction free surface the stress functions srr ¼ srz ¼ 0 at
r ¼ a,
szz ¼ srz ¼ 0
z ¼ h.
at
Eqs. (1)–(15) constitute mathematical formulation of the problem.
ð15Þ
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
32
3. Solution Introducing the finite Hankel transform over the variable r and its inverse transform defined as in [12]: Z a Tðan ; zÞ ¼ rJ 0 ðan rÞTðr; zÞ dr, (16) 0
Tðr; zÞ ¼
1 X n¼1
! 2J 0 ðan rÞ Tðan ; zÞ; a2 J 20 ðan aÞ
(17)
where a1, a2, y are roots of the transcendental equation J 1 ðaaÞ ¼ 0, Jn(x) is Bessel function of the first kind of order n. This transform satisfies the relations 2 q T 1 qT H þ ¼ a2n Tðam ; zÞ, qr2 r qr
(18)
(19)
and H
q2 T d2 T ¼ 2. 2 qz dz
(20)
On applying the finite Hankel transform defined in Eq. (16) to Eq. (2), one obtains d2 T a2n T ¼ 0, dz2 where T is the Hankel transform of T. On solving Eq. (21) under the condition given in Eqs. (4) and (5), one obtains 1 X f ðan Þ cosh½an ðz þ hÞ T¼ . cosh½an ðx þ hÞ n¼1 On applying inverse Hankel transform defined in Eq. (17), one obtains " # X 2 1 f ðan ÞJ 0 ðan rÞ cosh½an ðz þ hÞ T¼ , a2 n¼1 cosh½an ðx þ hÞ J 20 ðan aÞ
(21)
(22)
(23)
where f ðan Þ is the Hankel transform of f(r). The unknown temperature g(r) can be obtained by substituting z ¼ h in Eq. (23) as " # X 2 1 f ðan ÞJ 0 ðan rÞ coshð2an hÞ gð r Þ ¼ . (24) a2 n¼1 cosh½an ðx þ hÞ J 20 ðan aÞ Since initial temperature Ti ¼ 0, the temperature change t ¼ T.
(25)
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
33
Now, suitable form of M satisfying Eq. (9) is given by " # X 2K 1 f ðan ÞJ 0 ðan rÞ Bn sinh½an ðz þ hÞþC n an ½z þ h cosh½an ðz þ hÞ , M¼ 2 2 a J 0 ðan aÞ n¼1 (26) where Bn and Cn are arbitrary functions. Assuming displacement function f(r, z) as 1 X ðz þ hÞ sinh½an ðz þ hÞ fðr; zÞ ¼ Dn J 0 ðan rÞ , cosh½an ðx þ hÞ n¼1
(27)
and using f in Eq. (1), one has Dn ¼
Kf ðan Þ . 2 n J 0 ðan aÞ
a2 a
Thus Eq. (27) becomes # (" X ) K 1 f ðan ÞJ 0 ðan rÞ ðz þ hÞ sinh½an ðz þ hÞ : fðr; zÞ ¼ a2 n¼1 an cosh½an ðx þ hÞ J 20 ðan aÞ
(28)
Now using Eqs. (23), (25), (26) and (28) in Eqs. (7), (8) and (11)–(14), one obtains the expressions for displacements and stresses, respectively, as X K 1 f ðan ÞJ 1 ðan rÞ ðz þ hÞ sinh½an ðz þ hÞ ur ¼ þ Bn a2n cosh½an ðz þ hÞ a2 n¼1 J 20 ðan aÞ cosh½an ðx þ hÞ 2 þ C n an ½cosh½an ðz þ hÞ þ an ½z þ h sinh½an ðz þ hÞ , ð29Þ " # X K 1 f¯ ðan ÞJ 0 ðan rÞ ½sinh ½an ðz þ hÞ þ an ½z þ h cosh ½an ðz þ hÞ uz ¼ a2 n¼1 an cosh ½an ðx þ hÞ J 20 ðan aÞ Bn a2n sinh ½an ðz þ hÞ þ C n a2n ½2ð1 2vÞ sinh ½an ðz þ hÞ ) an ½z þ h cosh½an ðz þ hÞ ,
ð30Þ
#8 1" 2KG X f ðan Þ < J 1 ðan rÞ ðz þ hÞ sinh½an ðz þ hÞ srr ¼ a J ða rÞ 2 : n 0 n a2 r cosh½an ðx þ hÞ n¼1 J 0 ðan aÞ cosh½an ðz þ hÞ J 1 ðan rÞ 2 2J 0 ðan rÞ cosh½an ðz þ hÞ þ Bn an an J 0 ðan rÞ cosh½an ðx þ hÞ r 2 39 2van J 0 ðan rÞ cosh½an ðz þ hÞ > = 7 26 þ C n an 4 J 1 ðan rÞ 5 ½cosh½an ðz þ hÞ þ an ½z þ h sinh½an ðz þ hÞ > þ an J 0 ðan rÞ ; r ð31Þ
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
34
syy
# 1" ¯ f ðan Þ 2KG X J 1 ðan rÞ ðz þ hÞ sinh ½an ðz þ hÞ ¼ 2 a2 r cosh ½an ðx þ hÞ n¼1 J 0 ðan aÞ cosh ½an ðz þ hÞ Bn a2n J 1 ðan rÞ cosh ½an ðz þ hÞ 2J 0 ðan rÞ þ cosh ½an ðx þ hÞ r 2 39 2van J 0 ðan rÞ cosh½an ðz þ hÞ > = 7 26 þ C n an 4 J 1 ðan rÞ 5 ½cosh½an ðz þ hÞ þ an ½z þ h sinh½an ðz þ hÞ > þ ; r
szz ¼
ð32Þ
# 1 "¯ 2KG X f ðan ÞJ 0 ðan rÞ ðz þ hÞan sinh ½an ðz þ hÞ Bn a3n cosh ½an ðz þ hÞ 2 a2 cosh ½a ðx þ hÞ J ða aÞ n 0 n n¼1 )
þ C n a3n ½ð1 2vÞ cosh½an ðz þ hÞan ðz þ hÞ sinh ½an ðz þ hÞ
ð33Þ
and # 1" 2KG X f ðan ÞJ 1 ðan rÞ sinh½an ðz þ hÞ an ðz þ hÞ cosh½an ðz þ hÞ srz ¼ a2 cosh½an ðx þ hÞ J 20 ðan aÞ n¼1 þ Bn a3n sinh½an ðz þ hÞ þ
C n a3n ½2v
sinh½an ðz þ hÞþan ðz þ hÞ cosh½an ðz þ hÞ .
ð34Þ
Now in order to satisfy Eq. (15) solving Eqs. (31), (33) and (34) for Bn and Cn one obtains, Bn ¼
ð1 2nÞ , a3n cosh½an ðx þ hÞ
(35)
Cn ¼
1 . a3n cosh½an ðx þ hÞ
(36)
and
Using these values of Bn and Cn in Eqs. (29)–(34) one obtain the expressions for displacements and stresses as " # X 2K 1 f ðan ÞJ 1 ðan rÞ ð1 nÞ cosh½an ðz þ hÞ ur ¼ , (37) a2 n¼1 an cosh½an ðx þ hÞ J 20 ðan aÞ uz ¼
# 1" 2K X f ðan ÞJ 0 ðan rÞ ð1 nÞ sinh½an ðz þ hÞ ; a2 n¼1 an cosh½an ðx þ hÞ J 20 ðan aÞ
# 1" 4KG X f ðan ÞJ 1 ðan rÞ ð1 nÞ cosh½an ðz þ hÞ srr ¼ , a2 an cosh½an ðx þ hÞ rJ 20 ðan aÞ n¼1
(38)
(39)
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
syy ¼
# 1" 4KG X f ðan Þ J 1 ðan rÞ ð1 nÞ cosh½an ðz þ hÞ an J 0 ðan rÞ , 2 a2 r an cosh½an ðx þ hÞ n¼1 J 0 ðan aÞ
35
(40)
szz ¼ 0,
(41)
srz ¼ 0.
(42)
and
4. Numerical calculations Set
2 f ðrÞ ¼ r2 a2 . Applying finite Hankel transform as defined in Eq. (16) to Eq. (43), one obtains Z a
2 f ðan Þ ¼ r r2 a2 J 0 ðan rÞ dr, 0
8a 8 a2 a2n J 1 ðan aÞ 4aan J 0 ðan aÞ f ðan Þ ¼ . a5n
(43)
ð44Þ
The numerical calculations have been carried out for steel (SN 50C) plate with the parameters a ¼ 1 m, h ¼ 0.2 m, thermal diffusivity k ¼ 15.9 106 m2 s1 and Poisson ratio n ¼ 0.281 with a1 ¼ 3.8317, a2 ¼ 7.0156, a3 ¼ 10.1735, a4 ¼ 13.3237, a5 ¼ 16.470, a6 ¼ 19.6159, a7 ¼ 22.7601, a8 ¼ 25.9037, a9 ¼ 29.0468, a10 ¼ 32.18 being the positive roots of transcendental equation J1(a a) ¼ 0. For convenience setting A ¼ 16=102 a; B ¼ 16K=102 a; C ¼ 32GK=102 a in the expressions (24), (37)–(40). The numerical expressions for unknown temperature, displacement and stress components are obtained by Eqs. (24) and (37)–(42). In order to examine the influence of unknown temperature on the upper surface of circular plate, one performes the numerical calculations z ¼ h/2, r ¼ 0; 0:2; 0:4; 0:6; 0:8 and 1and x ¼ 0:2; 0:1; 0 and 0:1. Numerical variations in radial directions are shown in the figures with the help of computer programme. 5. Concluding remarks In this paper a circular plate is considered which is subjected to arbitrary known interior temperature and determined the expressions for unknown temperature, displacements and stress functions, due to unknown temperature determined. As a special case a mathematical model is constructed for f ðrÞ ¼ ðr2 a2 Þ2 and numerical calculations are performed. The thermoelastic behaviors such as temperature, displacements and stresses are examined with the help of arbitrary known interior temperature. Fig. 1 shows that the unknown temperature increases with the thickness of the circular plate. Fig. 2 shows that the radial displacement increases with the thickness of the circular plate and Fig. 3 shows that the axial displacement decreases with the thickness of the circular plate. Fig. 3, Radial stress function srr is zero at center (r ¼ 0) and at circular boundary (r ¼ a). Also, it increases with the thickness of the circular plate, and it develops
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V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
400 300 a
200 100
b,c & d
0 -100
0
0.2
0.4
0.6
0.8
1
r
25 20 15 10 5 0 -5 0 -10
b
c d
0.2
0.4
0.6
0.8
1
r Fig. 1. The unknown temperature g(r)/A in radial direction (with its magnifying image). (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
2.5 a
2
b
1.5
c
1 d
0.5 0 0 4 2 0 -2 0 -4 -6 -8 -10
0.2
0.4
r
0.6
0.8
0.6
0.8
1
c&d
0.2
0.4
b
1
a
r
Fig. 2. The displacement functions ur/B and uz/B in radial direction. (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
the tensile stresses in radial direction. Fig. 4 shows that the stress function syy increases with the thickness of the circular plate. It is negligible for small thickness. Also it develops the tensile stresses in radial direction.
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
12 10 8 6 4 2 0 -2 0
37
a
b
c
d
0.2
0.4
0.6
r
0.8
1
Fig. 3. The radial stress function srr/C in radial direction. (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
400 300
a
200 100
b,c & d
0 -100
0
25 20 15 10 5 0 -5 0 -10
0.2
0.4
0.6
0.8
1
r b
c d
0.2
0.4
0.6
0.8
1
r
Fig. 4. The radial stress function syy/C in radial direction (with its magnifying image). (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
It means we may find out that displacement and stress components occuring near heat source. With the temperature increases the circular plate will tend to expand in radial as well as in axial direction. In the plane state of stress the stress components szz and srz are zero. Also from the figure of displacement it can be observed that displacement occurs around the center towards downward direction. So it may be concluded that due to unknown temperature the circular plate expands in axial direction and bends concavely at the center. This expansion is inversely proportional to thickness of circular plate. The results obtained here are more useful in engineering problems particularly in the determination of state of strain in thick annular disc. Also any particular case of special interest can be derived by assigning suitable values to the parameters and function in the expressions (24) and (37)–(42).
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Acknowledgment The authors are thankful to University Grants Commission, New Delhi for providing the partial financial assistance under major research project scheme. References [1] N. Noda, F. Ashida, T. Tsuji, An inverse transient thermoelastic problem of a transversely isotropic body, J. Applied Mech. (Trans. ASME Ser. E) 56 (4) (1989) 791–797. [2] F. Ashida, S. Sakata, T.R. Tauchert, Y. Yamashita, Inverse transient thermoelastic problem for a composite circular disc, Journal of Therm. Stresses 25 (2002) 431–455. [3] K. Grysa, Z. Kozlowski, One dimensional problem of temperature and heat flux determination at the surfaces of a thermoelastic slab. Part II: the numerical analysis, Nucl. Eng. Des. 74 (1982) 15–24. [4] K.C. Sabherwal, An inverse problem of transient heat conduction, Indian J. Pure Appl. Phys. 3E (10) (1965) 397–398. [5] K.C. Deshmukh, P.C. Wankhede, Inverse thermoelastic problem in an anisotropic cylinder, J. Indian Math. Soc. 68 (1–4) (2001) 75–80. [6] K.C. Deshmukh, P.C. Wankhede, An inverse transient problem of quasi-static thermal deflection of a thin clamped circular plate, Bull. Pure Appl. Sci. 17E (1) (1998). [7] K.C. Deshmukh, P.C. Wankhede, An axisymmetric inverse steady state problem of thermoelastic deformation of a finite length hollow cylinder, Far East J. Appl. Math. 1 (3) (1997). [8] K.C. Deshmukh, P.C. Wankhede, An inverse quasi-static transient thermoelastic problem in a thin annular disc, Far East J. Appl. Math. 2 (2) (1998) 117–124. [9] P.C. Wankhede, On the quasi static thermal stresses in a circular plate, Indian J. Pure Appl. Math. 13 (11) (1982) 1273–1277. [10] V.S. Gogulwar, K.C. Deshmukh, An inverse quasi-static thermal stresses in an annular disc, Proceeding of the ICADS, Narosa Publishing House, 2002. [11] N. Noda, R.B. Hetnarski, Y. Tanigawa, Thermal Stresses, second ed., Taylor & Francis, New York, 2003, pp. 259–261. [12] I.N. Sneddon, The Use of Integral Transform, McGraw-Hill, New York, 1972, pp. 235–238.
Journal of the Franklin Institute 345 (2008) 29–38 www.elsevier.com/locate/jfranklin
An inverse quasi-static steady-state thermal stresses in a thick circular plate V.S. Kulkarnia, K.C. Deshmukhb, a
Department of Mathematics, Government College of Engineering, Chandrapur 442 401, Maharashtra, India b Post-Graduate Department of Mathematics, Nagpur University, Nagpur 440 010, Maharashtra, India Received 6 November 2006; received in revised form 1 June 2007; accepted 11 June 2007
Abstract The present paper deals with the determination of unknown temperature and thermal stresses on the upper surface of a thick circular plate. A thick circular plate is subjected to arbitrary known interior temperature under steady state. The fixed circular edge and lower surface of the circular plate are thermally insulated. The governing heat conduction equation has been solved by using Hankel transform methods. The results are obtained in series form in terms of Bessel’s functions. The results for displacement and stresses have been computed numerically and illustrated graphically. r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Steady state; Thermoelastic problem; Inverse; Thermal stresses
1. Introduction During the second half of the twentieth century, nonisothermal problems of the theory of elasticity became increasingly important. This is due to their wide application in diverse fields. The high velocities of modern aircraft give rise to aerodynamic heating, which produces intense thermal stresses that reduce the strength of the aircraft structure. Noda et al. [1] discussed an analytical method for an inverse problem of threedimensional transient thermoelasticity in a transversely isotropic solid by integral transform technique with newly designed potential function and illustrated practical applicability of the method in engineering problem. Ashida et al. [2] studied the inverse Corresponding author.
E-mail addresses: [email protected] (V.S. Kulkarni), [email protected] (K.C. Deshmukh). 0016-0032/$32.00 r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2007.06.003
ARTICLE IN PRESS 30
V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
transient thermoelastic problem for a composite circular disc constructed of transversely isotropic layer. Grysa and Kozlowski [3] investigated an inverse one-dimensional transient thermoelastic problem and obtained the temperature and heat flux on the surface of an isotropic infinite slab. Sabherwal [4] studied an inverse problem in heat conduction. Deshmukh and Wankhede [5] solved the inverse thermoelastic problem in an anisotropic cylinder and determined the temperature and thermal stresses on the outer curved surface of a cylinder. Also, Deshmukh and Wankhede [6–8] studied inverse problem on hollow cylinders, circular plates and annular disc by applying integral transform technique and illustrated them numerically. Wankhede [9] determined quasi-static thermal stresses in thin circular plate. Recently Gogulwar and Deshmukh [10] solved the inverse problem of thermal stresses in a thin annular disc. This paper deals with the realistic problem of inverse quasi-static steady-state thermal stresses in a thick circular plate, which is subjected to arbitrary interior temperature. The fixed circular edge and lower surface of the circular plate are thermally insulated. The unknown temperature and thermal stresses on the upper surface of thick circular plate is required to be determined. Due to unknown temperature on the upper surface of the thick circular plate, it expands in axial direction and bends concavely at the center. This expansion is inversely proportional to thickness of circular plate, which is a new and novel contribution of this paper. The results presented here are more useful in engineering problem particularly in the determination of the state of strain in circular plate constituting foundations of containers for hot gases or liquids, in the foundations for furnaces, etc. 2. Formulation of the problem Consider a circular plate of thickness 2h occupying space D defined by 0prpa, hpzph. Let the plate be subjected to arbitrary known interior temperature f(r) within region hozoh, with lower surface z ¼ h and circular surface r ¼ a thermally insulated. Under these more realistic prescribed conditions, the unknown temperature g(r) which is at the upper surface of plate z ¼ h and quasi-static thermal stresses due to unknown temperature g(r) are required to be determined. The differential equation governing the displacement potential function f(r, z) is given in [11] as q2 f 1 qf q2 f þ þ ¼ Kt, qr2 r qr qz2
(1)
where K is the restraint coefficient and temperature change t ¼ T T i , Ti is the initial temperature. Displacement function f is known as Goodier’s thermoelastic displacement potential. The steady-state temperature of the disc satisfies the heat conduction equation q2 T 1 qT q2 T þ 2 ¼ 0, þ qr2 r qr qz
(2)
with the conditions qT ¼0 qr
at r ¼ a;
hpzph,
(3)
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
qT ¼ 0 at qz
z ¼ h;
0prpa,
T ¼ f ðrÞ ðknownÞ at z ¼ x;
31
(4)
hpxoh; 0prpa,
(5)
and T ¼ gðrÞ
ðunknownÞ at z ¼ h;
0prpa.
(6)
The displacement functions in the cylindrical coordinate system are represented by the Michell’s function defined in [11] as ur ¼
qf q2 M , qr qr dz
(7)
uz ¼
qf q2 M þ 2ð1 nÞr2 M 2 . qz qz
(8)
The Michell’s function M must satisfy r2 r2 M ¼ 0,
(9)
where r2 ¼
q2 1 q q2 þ þ 2. 2 r qr qz qr
(10)
The component of the stresses are represented by the thermoelastic displacement potential f and Michell’s function M as 2 q f q q2 M 2 srr ¼ 2G vr Kt þ M , (11) qr2 qz qr2
1 qf q 1 qM Kt þ vr2 M , r qr qz r qr
(12)
2 q f q q2 M 2 ð2 vÞr Kt þ M qz2 qz qz2
(13)
syy ¼ 2G
szz ¼ 2G and
q2 f q q2 M 2 þ ð1 vÞr M 2 srz ¼ 2G , qrqz qr qz
(14)
where G and n are the shear modulus and Poisson’s ratio, respectively. For traction free surface the stress functions srr ¼ srz ¼ 0 at
r ¼ a,
szz ¼ srz ¼ 0
z ¼ h.
at
Eqs. (1)–(15) constitute mathematical formulation of the problem.
ð15Þ
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
32
3. Solution Introducing the finite Hankel transform over the variable r and its inverse transform defined as in [12]: Z a Tðan ; zÞ ¼ rJ 0 ðan rÞTðr; zÞ dr, (16) 0
Tðr; zÞ ¼
1 X n¼1
! 2J 0 ðan rÞ Tðan ; zÞ; a2 J 20 ðan aÞ
(17)
where a1, a2, y are roots of the transcendental equation J 1 ðaaÞ ¼ 0, Jn(x) is Bessel function of the first kind of order n. This transform satisfies the relations 2 q T 1 qT H þ ¼ a2n Tðam ; zÞ, qr2 r qr
(18)
(19)
and H
q2 T d2 T ¼ 2. 2 qz dz
(20)
On applying the finite Hankel transform defined in Eq. (16) to Eq. (2), one obtains d2 T a2n T ¼ 0, dz2 where T is the Hankel transform of T. On solving Eq. (21) under the condition given in Eqs. (4) and (5), one obtains 1 X f ðan Þ cosh½an ðz þ hÞ T¼ . cosh½an ðx þ hÞ n¼1 On applying inverse Hankel transform defined in Eq. (17), one obtains " # X 2 1 f ðan ÞJ 0 ðan rÞ cosh½an ðz þ hÞ T¼ , a2 n¼1 cosh½an ðx þ hÞ J 20 ðan aÞ
(21)
(22)
(23)
where f ðan Þ is the Hankel transform of f(r). The unknown temperature g(r) can be obtained by substituting z ¼ h in Eq. (23) as " # X 2 1 f ðan ÞJ 0 ðan rÞ coshð2an hÞ gð r Þ ¼ . (24) a2 n¼1 cosh½an ðx þ hÞ J 20 ðan aÞ Since initial temperature Ti ¼ 0, the temperature change t ¼ T.
(25)
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33
Now, suitable form of M satisfying Eq. (9) is given by " # X 2K 1 f ðan ÞJ 0 ðan rÞ Bn sinh½an ðz þ hÞþC n an ½z þ h cosh½an ðz þ hÞ , M¼ 2 2 a J 0 ðan aÞ n¼1 (26) where Bn and Cn are arbitrary functions. Assuming displacement function f(r, z) as 1 X ðz þ hÞ sinh½an ðz þ hÞ fðr; zÞ ¼ Dn J 0 ðan rÞ , cosh½an ðx þ hÞ n¼1
(27)
and using f in Eq. (1), one has Dn ¼
Kf ðan Þ . 2 n J 0 ðan aÞ
a2 a
Thus Eq. (27) becomes # (" X ) K 1 f ðan ÞJ 0 ðan rÞ ðz þ hÞ sinh½an ðz þ hÞ : fðr; zÞ ¼ a2 n¼1 an cosh½an ðx þ hÞ J 20 ðan aÞ
(28)
Now using Eqs. (23), (25), (26) and (28) in Eqs. (7), (8) and (11)–(14), one obtains the expressions for displacements and stresses, respectively, as X K 1 f ðan ÞJ 1 ðan rÞ ðz þ hÞ sinh½an ðz þ hÞ ur ¼ þ Bn a2n cosh½an ðz þ hÞ a2 n¼1 J 20 ðan aÞ cosh½an ðx þ hÞ 2 þ C n an ½cosh½an ðz þ hÞ þ an ½z þ h sinh½an ðz þ hÞ , ð29Þ " # X K 1 f¯ ðan ÞJ 0 ðan rÞ ½sinh ½an ðz þ hÞ þ an ½z þ h cosh ½an ðz þ hÞ uz ¼ a2 n¼1 an cosh ½an ðx þ hÞ J 20 ðan aÞ Bn a2n sinh ½an ðz þ hÞ þ C n a2n ½2ð1 2vÞ sinh ½an ðz þ hÞ ) an ½z þ h cosh½an ðz þ hÞ ,
ð30Þ
#8 1" 2KG X f ðan Þ < J 1 ðan rÞ ðz þ hÞ sinh½an ðz þ hÞ srr ¼ a J ða rÞ 2 : n 0 n a2 r cosh½an ðx þ hÞ n¼1 J 0 ðan aÞ cosh½an ðz þ hÞ J 1 ðan rÞ 2 2J 0 ðan rÞ cosh½an ðz þ hÞ þ Bn an an J 0 ðan rÞ cosh½an ðx þ hÞ r 2 39 2van J 0 ðan rÞ cosh½an ðz þ hÞ > = 7 26 þ C n an 4 J 1 ðan rÞ 5 ½cosh½an ðz þ hÞ þ an ½z þ h sinh½an ðz þ hÞ > þ an J 0 ðan rÞ ; r ð31Þ
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
34
syy
# 1" ¯ f ðan Þ 2KG X J 1 ðan rÞ ðz þ hÞ sinh ½an ðz þ hÞ ¼ 2 a2 r cosh ½an ðx þ hÞ n¼1 J 0 ðan aÞ cosh ½an ðz þ hÞ Bn a2n J 1 ðan rÞ cosh ½an ðz þ hÞ 2J 0 ðan rÞ þ cosh ½an ðx þ hÞ r 2 39 2van J 0 ðan rÞ cosh½an ðz þ hÞ > = 7 26 þ C n an 4 J 1 ðan rÞ 5 ½cosh½an ðz þ hÞ þ an ½z þ h sinh½an ðz þ hÞ > þ ; r
szz ¼
ð32Þ
# 1 "¯ 2KG X f ðan ÞJ 0 ðan rÞ ðz þ hÞan sinh ½an ðz þ hÞ Bn a3n cosh ½an ðz þ hÞ 2 a2 cosh ½a ðx þ hÞ J ða aÞ n 0 n n¼1 )
þ C n a3n ½ð1 2vÞ cosh½an ðz þ hÞan ðz þ hÞ sinh ½an ðz þ hÞ
ð33Þ
and # 1" 2KG X f ðan ÞJ 1 ðan rÞ sinh½an ðz þ hÞ an ðz þ hÞ cosh½an ðz þ hÞ srz ¼ a2 cosh½an ðx þ hÞ J 20 ðan aÞ n¼1 þ Bn a3n sinh½an ðz þ hÞ þ
C n a3n ½2v
sinh½an ðz þ hÞþan ðz þ hÞ cosh½an ðz þ hÞ .
ð34Þ
Now in order to satisfy Eq. (15) solving Eqs. (31), (33) and (34) for Bn and Cn one obtains, Bn ¼
ð1 2nÞ , a3n cosh½an ðx þ hÞ
(35)
Cn ¼
1 . a3n cosh½an ðx þ hÞ
(36)
and
Using these values of Bn and Cn in Eqs. (29)–(34) one obtain the expressions for displacements and stresses as " # X 2K 1 f ðan ÞJ 1 ðan rÞ ð1 nÞ cosh½an ðz þ hÞ ur ¼ , (37) a2 n¼1 an cosh½an ðx þ hÞ J 20 ðan aÞ uz ¼
# 1" 2K X f ðan ÞJ 0 ðan rÞ ð1 nÞ sinh½an ðz þ hÞ ; a2 n¼1 an cosh½an ðx þ hÞ J 20 ðan aÞ
# 1" 4KG X f ðan ÞJ 1 ðan rÞ ð1 nÞ cosh½an ðz þ hÞ srr ¼ , a2 an cosh½an ðx þ hÞ rJ 20 ðan aÞ n¼1
(38)
(39)
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
syy ¼
# 1" 4KG X f ðan Þ J 1 ðan rÞ ð1 nÞ cosh½an ðz þ hÞ an J 0 ðan rÞ , 2 a2 r an cosh½an ðx þ hÞ n¼1 J 0 ðan aÞ
35
(40)
szz ¼ 0,
(41)
srz ¼ 0.
(42)
and
4. Numerical calculations Set
2 f ðrÞ ¼ r2 a2 . Applying finite Hankel transform as defined in Eq. (16) to Eq. (43), one obtains Z a
2 f ðan Þ ¼ r r2 a2 J 0 ðan rÞ dr, 0
8a 8 a2 a2n J 1 ðan aÞ 4aan J 0 ðan aÞ f ðan Þ ¼ . a5n
(43)
ð44Þ
The numerical calculations have been carried out for steel (SN 50C) plate with the parameters a ¼ 1 m, h ¼ 0.2 m, thermal diffusivity k ¼ 15.9 106 m2 s1 and Poisson ratio n ¼ 0.281 with a1 ¼ 3.8317, a2 ¼ 7.0156, a3 ¼ 10.1735, a4 ¼ 13.3237, a5 ¼ 16.470, a6 ¼ 19.6159, a7 ¼ 22.7601, a8 ¼ 25.9037, a9 ¼ 29.0468, a10 ¼ 32.18 being the positive roots of transcendental equation J1(a a) ¼ 0. For convenience setting A ¼ 16=102 a; B ¼ 16K=102 a; C ¼ 32GK=102 a in the expressions (24), (37)–(40). The numerical expressions for unknown temperature, displacement and stress components are obtained by Eqs. (24) and (37)–(42). In order to examine the influence of unknown temperature on the upper surface of circular plate, one performes the numerical calculations z ¼ h/2, r ¼ 0; 0:2; 0:4; 0:6; 0:8 and 1and x ¼ 0:2; 0:1; 0 and 0:1. Numerical variations in radial directions are shown in the figures with the help of computer programme. 5. Concluding remarks In this paper a circular plate is considered which is subjected to arbitrary known interior temperature and determined the expressions for unknown temperature, displacements and stress functions, due to unknown temperature determined. As a special case a mathematical model is constructed for f ðrÞ ¼ ðr2 a2 Þ2 and numerical calculations are performed. The thermoelastic behaviors such as temperature, displacements and stresses are examined with the help of arbitrary known interior temperature. Fig. 1 shows that the unknown temperature increases with the thickness of the circular plate. Fig. 2 shows that the radial displacement increases with the thickness of the circular plate and Fig. 3 shows that the axial displacement decreases with the thickness of the circular plate. Fig. 3, Radial stress function srr is zero at center (r ¼ 0) and at circular boundary (r ¼ a). Also, it increases with the thickness of the circular plate, and it develops
ARTICLE IN PRESS 36
V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
400 300 a
200 100
b,c & d
0 -100
0
0.2
0.4
0.6
0.8
1
r
25 20 15 10 5 0 -5 0 -10
b
c d
0.2
0.4
0.6
0.8
1
r Fig. 1. The unknown temperature g(r)/A in radial direction (with its magnifying image). (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
2.5 a
2
b
1.5
c
1 d
0.5 0 0 4 2 0 -2 0 -4 -6 -8 -10
0.2
0.4
r
0.6
0.8
0.6
0.8
1
c&d
0.2
0.4
b
1
a
r
Fig. 2. The displacement functions ur/B and uz/B in radial direction. (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
the tensile stresses in radial direction. Fig. 4 shows that the stress function syy increases with the thickness of the circular plate. It is negligible for small thickness. Also it develops the tensile stresses in radial direction.
ARTICLE IN PRESS V.S. Kulkarni, K.C. Deshmukh / Journal of the Franklin Institute 345 (2008) 29–38
12 10 8 6 4 2 0 -2 0
37
a
b
c
d
0.2
0.4
0.6
r
0.8
1
Fig. 3. The radial stress function srr/C in radial direction. (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
400 300
a
200 100
b,c & d
0 -100
0
25 20 15 10 5 0 -5 0 -10
0.2
0.4
0.6
0.8
1
r b
c d
0.2
0.4
0.6
0.8
1
r
Fig. 4. The radial stress function syy/C in radial direction (with its magnifying image). (a) x ¼ 0.2, (b) x ¼ 0.1, (c) x ¼ 0, and (d) x ¼ 0.1.
It means we may find out that displacement and stress components occuring near heat source. With the temperature increases the circular plate will tend to expand in radial as well as in axial direction. In the plane state of stress the stress components szz and srz are zero. Also from the figure of displacement it can be observed that displacement occurs around the center towards downward direction. So it may be concluded that due to unknown temperature the circular plate expands in axial direction and bends concavely at the center. This expansion is inversely proportional to thickness of circular plate. The results obtained here are more useful in engineering problems particularly in the determination of state of strain in thick annular disc. Also any particular case of special interest can be derived by assigning suitable values to the parameters and function in the expressions (24) and (37)–(42).
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Acknowledgment The authors are thankful to University Grants Commission, New Delhi for providing the partial financial assistance under major research project scheme. References [1] N. Noda, F. Ashida, T. Tsuji, An inverse transient thermoelastic problem of a transversely isotropic body, J. Applied Mech. (Trans. ASME Ser. E) 56 (4) (1989) 791–797. [2] F. Ashida, S. Sakata, T.R. Tauchert, Y. Yamashita, Inverse transient thermoelastic problem for a composite circular disc, Journal of Therm. Stresses 25 (2002) 431–455. [3] K. Grysa, Z. Kozlowski, One dimensional problem of temperature and heat flux determination at the surfaces of a thermoelastic slab. Part II: the numerical analysis, Nucl. Eng. Des. 74 (1982) 15–24. [4] K.C. Sabherwal, An inverse problem of transient heat conduction, Indian J. Pure Appl. Phys. 3E (10) (1965) 397–398. [5] K.C. Deshmukh, P.C. Wankhede, Inverse thermoelastic problem in an anisotropic cylinder, J. Indian Math. Soc. 68 (1–4) (2001) 75–80. [6] K.C. Deshmukh, P.C. Wankhede, An inverse transient problem of quasi-static thermal deflection of a thin clamped circular plate, Bull. Pure Appl. Sci. 17E (1) (1998). [7] K.C. Deshmukh, P.C. Wankhede, An axisymmetric inverse steady state problem of thermoelastic deformation of a finite length hollow cylinder, Far East J. Appl. Math. 1 (3) (1997). [8] K.C. Deshmukh, P.C. Wankhede, An inverse quasi-static transient thermoelastic problem in a thin annular disc, Far East J. Appl. Math. 2 (2) (1998) 117–124. [9] P.C. Wankhede, On the quasi static thermal stresses in a circular plate, Indian J. Pure Appl. Math. 13 (11) (1982) 1273–1277. [10] V.S. Gogulwar, K.C. Deshmukh, An inverse quasi-static thermal stresses in an annular disc, Proceeding of the ICADS, Narosa Publishing House, 2002. [11] N. Noda, R.B. Hetnarski, Y. Tanigawa, Thermal Stresses, second ed., Taylor & Francis, New York, 2003, pp. 259–261. [12] I.N. Sneddon, The Use of Integral Transform, McGraw-Hill, New York, 1972, pp. 235–238.