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A numerical solution of magneto-thermoelastic problem in non-homogeneous isotropic cylinder by the finite-difference method
Applied Mathematical Modelling 31 (2007) 1662–1670 www.elsevier.com/locate/apm
A numerical solution of magneto-thermoelastic problem in non-homogeneous isotropic cylinder by the finite-difference method M.R. Abd-El-Salam a, A.M. Abd-Alla b, Hany A. Hosham
c,*
a
Department of Mathematics, Faculty of Science, South-Valley University, Sohag, Egypt Department of Mathematics, Faculty of Science, Taif University, Kingdom of Saudi Arabia c Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
b
Received 1 December 2005; received in revised form 1 April 2006; accepted 11 May 2006 Available online 7 August 2006
Abstract This paper presents the development of the magneto-thermoelastic problem in non-homogeneous isotropic cylinder in a primary magnetic field when the curved surface of the cylinder subject to certain boundary conditions. The governing coupled linear partial differential equations in the hyperbolic-type have been solved numerically using the finite-difference method. Graphical results for the temperature, displacement and components of stresses are illustrated and discussed for copper-like material. The results indicate that the effects of inhomogeneity and magnetic field are very pronounced. Some more interesting particular cases have also been discussed. 2006 Elsevier Inc. All rights reserved. Keywords: Magneto-thermoelastic; Coupled problems; Finite-difference method
1. Introduction The generalized theories of thermoelasticity, which admit the finite speed of thermal signal, have been the center of interest of active research during last three decades. The theory of generalized thermoelasticity was proposed by Lord and Shulman [1] and Green and Lindsay [2] hereafter called LS and GL theories, respectively. These theories have been developed by introducing one or two relaxation time in the thermoelastic process, with an aim to eliminate the paradox of infinite speed for the propagation of thermal signals. The two theories are structurally different form one another, and one cannot be obtained as a particular case of the other. Also, these theories involved a hyperbolic-type heat equation. These generalized theories were motivated by experiments exhibiting the actual occurrence of wave type heat transport and were considered to be more realistic than the conventional theory in dealing with practical problems involving very large heat fluxes and/or small time intervals, like those occurring in laser units and energy channels. *
Corresponding author. E-mail address: [email protected] (H.A. Hosham).
0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.05.009
M.R. Abd-El-Salam et al. / Applied Mathematical Modelling 31 (2007) 1662–1670
1663
The theory of magneto-thermoelasticity is concerned with the interacting effects of the applied magnetic field on the elastic and thermoelastic deformations of a solid body. This theory has aroused much interest in many industrial applications, particularly in nuclear devices, where there exists a primary magnetic field; various investigations are to be carried out by considering the interactions among magnetic, thermal and strain fields. Analysis of such problems also influence various applications in biomedical engineering as well as in different geomagnetic studies. The development of the interactions of an electromagnetic field, the thermal field, and the elastic field is a available in many studies see Green and Naghdi [3–5]. Among the authors who considered the generalized magneto-thermoelasticity equations is Suhubi [6] who studied magneto-thermo-viscoelastic interactions in a body having cylindrical geometry. Misra et al. [7,8] studied magneto-thermoelastic interactions in an infinite elastic medium with a cylindrical hole as well as in an aeolotropic solid cylinder subjected to a ramp-type heating. Dhaliwal et al. [9] discussed generalized magneto-thermoelastic wave in an infinite elastic solid with a cylindrical cavity. Mukhopadhyay and Roychoudhurj [10] discussed magneto-thermoelastic interactions in an infinite isotropic elastic cylinder subjected to a periodic loading. A review literature of hyperbolic thermoelasticity can be found in Chandrasekharaiah [11]. Recently, Mukhopadhyay [12] solved the thermoelastic interactions problem without energy dissipation in an unbounded medium, Abd-Alla et al. [13] have investigated the thermal stress in an infinite circular cylinder of orthotropic material. El-Naggar et al. [14] studied the thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Abd-Alla et al. [15] investigated the magneto-thermoelastic problem in nonhomogeneous isotropic cylinder. In the present paper, we have considered an magneto-thermoelastic problem which involved a hyperbolictype heat equation. We have discussed the generation of stress, temperature and magnetic field in an infinite isotropic elastic cylinder placed in a constant primary magnetic field. The conservation equation of an magneto-thermoelastic are first deduce in the non-homogeneous form. The governing equations are transformed into a dimensionless form and their solutions are then obtained by an finite-difference method. The outline of the coupled finite-difference solutions procedure, we compute the values of displacement and temperature on the first line by using initial conditions. Also, we use the central, forward and backward finite-difference approximations of a second-order accurate to eliminating fictitious points at initial and boundary conditions. Using initial and first lines we can compute the solutions at next time to arrive the final time. Finite-difference models seem to enjoy significant popularity for the ease with which they can be implemented. Numerical results are presented for the variation of the temperature, displacement, and stresses with the time and through the thickness of the cylinder. The effect of inhomogeneity and the magnetic field are very pronounced. 2. Formulation of the problem Let us consider a cylindrical coordinate system (r, h, z), for the axially symmetric problem, we have uh = 0, ur ¼ ur ðr; z; ^tÞ and uz ¼ uz ðr; z; ^tÞ, where ^t is the time. Furthermore, if only axisymmetric plane strain problem is considered, we have uh = uz = 0 and u ¼ ur ðr; ^tÞ. The strain–displacement relations are err ¼
ou ; or
u ehh ¼ ; r
ezz ¼ ezr ¼ erh ¼ ehz ¼ 0;
ð1Þ
where ui and eij are the displacement component and strain components, respectively. The stress–strain relations are ou u ou þ rrr ¼ k þ 2l c Tb ðr; ^tÞ; or r or ou ou u þ rhh ¼ k ð2Þ þ 2l c Tb ðr; ^tÞ; or or r ou u þ ; srr ¼ l0 H 20 or r
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where rij is the stress components, k and l are Lame’s constants, c = (3k + 2l)at, at is the coefficient of linear is the magnetic permeability and H0 is the primary thermal expansion, Tb ðr; ^tÞ is the absolute temperature, l magnetic field. Equation of motion in a linear homogeneous and isotropic elastic solid in the absence of body force is given by orrr 1 osrr o2 u þ ðrrr rhh Þ þ ¼q 2; r or or o^t
ð3Þ
where q is the mass density. The heat conduction equation when absence heat source is given by Mukhopadhyay [16] ! 2b o2 Tb o ou u 1 o Tb o T þ c 2 þ cT 0 2 þ ¼k ; or2 r or o^t o^t or r
ð4Þ
where T0 is the reference temperature, c = qcv is the specific heat at constant volume, k* is a material constant. In this study, we assume the non-homogeneous property of the material is characterized by k ¼ Lr2m ;
l ¼ vr2m ;
l0 ¼ v0 r2m ;
q ¼ q0 r2m ;
ð5Þ
0
where L, v, v and q0 are known constants. Substituting from Eq. (5) into Eq. (2) we obtain the stress–displacement relations are ou u ou 2m 0b þ rrr ¼ r L þ 2v c T ; or r or ou ou u 2m 0b þ rhh ¼ r L þ 2v c T ; ð6Þ or or r ou u þ ; srr ¼ r2m v0 H 20 or r where c 0 = (3L + 2v)at. Substituting from Eq. (6) into Eqs. (3) and (4) we obtain the equations of motion and heat conduction, respectively; ! b o2 u o2 u 1 ou u 2m o T þ ½2mðL þ v0 H 20 Þ g 2 c0 Tb þ q0 2 ¼ g 2 þ gð1 þ 2mÞ ; ð7Þ or r or r r or o^t ! 2 2b o2 Tb ou u 1 o Tb o T 0 2m o þ þ ¼k ; ð8Þ c 2 þ c T 0r or2 r or o^t o^t2 or r where g ¼ L þ 2v þ v0 H 20 . Now we introduce the non-dimensional quantities: k þ 2l r u; R¼ ; cT 0 a a rrr rhh ¼ ; rHH ¼ ; cT 0 cT 0
U¼ rRR
t t ¼ t; a
T ¼
sRR ¼
Tb ; T0
srr ; cT 0
ð9Þ
where t is a standard speed. The non-dimensional forms of Eqs. (7) and (8) are therefore obtained as o2 U o2 U 1 oU U 2m oT 2 þ a T þ ¼ c a þ a ; ð10Þ 1 2 3 2 p ot2 R oR R oR oR2 R 2 2 o2 T 1 oT oU U 2m o 2 o T þ ¼ cT þ eðaRÞ ; ð11Þ ot2 ot2 oR R oR2 R oR 2mðLþv0 H 2 Þg
02
c T0 g 0 where c2p ¼ Lþ2v , a1 ¼ Lþ2v , a2 ¼ gð1þ2mÞ , a3 ¼ , c2T ¼ ctk 2 , and e ¼ ðLþ2vÞc , and the non-dimensional forms Lþ2v Lþ2v qt2 of the stress–strain relation can be written are as follows:
M.R. Abd-El-Salam et al. / Applied Mathematical Modelling 31 (2007) 1662–1670
oU U þ k1 T ; rRR ¼ R oR R oU U 2m þ T ; rHH ¼ R k1 oR R v0 H 20 oU U þ sRR ¼ ; L þ 2v oR R
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2m
ð12Þ
k . where k1 ¼ kþ2l From preceding description, the initial condition may be expressed as
U ðR; 0Þ ¼
oU ðR; 0Þ ¼ 0; ot
T ðR; 0Þ ¼
oT ðR; 0Þ ¼ 0; ot
a 6 R 6 M;
ð13Þ
where a and M are inner and outer radii of the cylinder. The boundary conditions may be expressed as rRR ¼ 0; rRR ¼ 0;
T ¼ eXt ; at R ¼ a; t > 0; T ¼ 0; at R ¼ M; t > 0;
ð14Þ
where X is exponent of the decayed heat flux.
a
b
Fig. 1. Temperature distributions with m = 0.5 and H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5, and various values of X.
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3. Numerical solution of the problem The computational domain {(R, t) : R 2 [a, M], t 2 [0, 1)} is discretized into cells described by the node set (Ri, tn). The uniform mesh defined by Ri = a + ih; i = 0, 1, . . . , N; tn = nDt; n = 0, 1, . . . , K; h ¼ Ma is a mesh N width, Dt = tmax/K is the time step and tmax is the final time. Here, naturally, N and K are positive integers. In order to construct a numerical solution for the main problem (10)–(14) we introduce the explicit finitedifference scheme. Then the finite-difference equations corresponding to Eqs. (10) and (11) are as follows: a2 h n nþ1 n n1 U i ¼ 2U i U i þ q1 a1 ðU niþ1 2U ni þ U ni1 Þ þ ðU U ni1 Þ 2Ri iþ1 a3 h2 2mh2 n h n þ 2 U ni T i þ ðT iþ1 T ni1 Þ ; ð15Þ 2 Ri Ri h nþ1 n n1 n n n n n T i ¼ 2T i T i þ q2 ðT iþ1 2T i þ T i1 Þ þ ðT T i1 Þ 2Ri iþ1 where q1 ¼
eR2m e nþ1 n1 n n1 i ðU iþ1 2U niþ1 þ U iþ1 U nþ1 ðU nþ1 2U ni þ U n1 Þ; i1 þ 2U i1 U i1 Þ i Ri i 2h
c2p ðDtÞ2 h2
and q2 ¼
ð16Þ
c2T ðDtÞ2 . h2
a
b
Fig. 2. Radial displacement distributions with H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5, and various values of X.
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In order to start computation we require data on the two line t = 0, and t = Dt. The initial conditions (13) gives the initial line and values on the line t = Dt. Substituting the central difference approximation for the oU 0 oT 0 U 1 U 1 T 1 T 1 and T 1 from Eqs. (15) and (16) for n = 0, derivative, i.e. ot i ¼ i 2k i and oti ¼ i 2k i and eliminating U 1 i i we get the formulas to give the values on the first level. q1 a2 h 0 1 0 a1 ðU 0iþ1 2U 0i þ U 0i1 Þ þ Ui ¼ Ui þ ðU U 0i1 Þ 2Ri iþ1 2 a3 h2 2mh2 0 h 0 þ 2 U 0i T i þ ðT iþ1 T 0i1 Þ ; ð17Þ 2 Ri Ri q h T 1i ¼ T 0i þ 2 ðT 0iþ1 2T 0i þ T 0i1 Þ þ ðT 0iþ1 T 0i1 Þ 2Ri 2
eR2m e i ðU 1iþ1 U 0iþ1 U 1i1 þ U 0i1 Þ ðU 1i U 0i Þ: Ri 2h
ð18Þ
The boundary conditions at R = a, in the finite-difference form is U n1 U n1 Un þ k1 0 T n0 ¼ 0; 2h R0
T n0 ¼ eXt :
a
b
Fig. 3. Radial stress distributions with H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5 and various values of X.
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At R = a, t > 0 (i.e., i = 0), Eq. (15) becomes n a2 h 2hk1 U n0 nþ1 n n1 n n 2hk1 U 0 n n U 0 ¼ 2U 0 U 0 þ q1 a1 2U 1 2U 0 2T 0 þ þ 2T 0 2R0 R0 R0 a3 h2 2mh2 n h þ 2 U n0 T þ ð3T n0 þ 4T n1 T n2 Þ : R0 0 2 R0
ð19Þ
Also, the boundary conditions at R = M, in the finite-difference form is U nN þ1 U nN 1 Un þ k1 N T nN ¼ 0; 2h RN
T nN ¼ 0:
At R = M, t > 0 (i.e., i = N), Eq. (15) becomes 2hk1 U nN a2 h 2hk1 U nN n n n n U Nnþ1 ¼ 2U nN U n1 þ q a 2U 2U 2T þ 2T þ 1 1 N N 1 N N N 2RN RN R0 a3 h2 2mh2 n h þ 2 U nN T þ ð3T nN 4T nN 1 þ T nN 2 Þ : RN N 2 RN
ð20Þ
a
b
Fig. 4. Hoop stress distributions with H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5 and various values of X.
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The local truncation error is a second-order accurate in time and space O((Dt)2 + h2), and it’s tends to zero as Dt and h tend to zero. Hence the present method is compatible. 4. Numerical results and discussion In order to illustrate the problem, the copper material was chosen for purposes of numerical evaluations. The physical data which given as e = 0.0168, Ł = 1.387 · 1012 dyne/cm2, v = 0.448 · 1012 dyne/cm2, cp = 1.0, cT = 0.578 and v 0 = 1.0 Gauss/Oersted. Results are presented for cylinder with a = 1.0, M = 3.0. Fig. 1 shows the temperature variation in the non-homogeneous case when the primary magnetic field is constant H0 = 3.0 · 105, for various values of non-dimensional time, t (Fig. 1a) or various values of exponent decayed heat flux, X (Fig. 1b). It is noticed that the temperature decreases with increasing of t and X, and the solutions satisfied the boundary conditions. Fig. 2 presents the radial displacement distributions along radial direction R with the various values of the t and X. The values of displacement are also seen to increase with increasing of time and arrived maximum value at t = 0.5, but the displacement decrease with increasing exponent of the decayed heat flux. Also, we can find the difference between the non-homogeneous and homogeneous problems, beside, this difference increases with increase of t and decrease of X. Figs. 3 and 4 show the influence of the non-homogeneous of the material constants on the radial and hoop stresses with the same values of magnetic field H0 and the parameters as those in Fig. 2. Also, they show the difference among two cases namely non-homogeneous and homogeneous bodies.
a
b
c
Fig. 5. Radial displacement, radial stress and hoop stress distributions, respectively for various values of H0 with t = 0.5, and X = 1.0.
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The coupling effect resists the temperature rise, and the resisting effect increases with increasing time. The physical meaning of this phenomenon is that the coupling effect acts like a damper inside the material, reduces the rate of heat propagation, and thus slows down the temperature rise. The influence of the primary magnetic field and inhomogeneity on the radial displacement, radial stress and hoop stress are show in Fig. 5. we can see that, the solutions satisfied the boundary conditions, and the difference between homogeneous and non-homogeneous cases can be shown. The variation of temperature, displacement, and stresses are due to the effect of inertia and magnetic field. Also, the influence of the nonhomogeneous on displacement and stresses is very pronounced. 5. Conclusion In this paper, a second-order explicit finite-difference scheme has been described. The scheme solves the coupled hyperbolic equations on a uniform grid, and is quite efficient for computation thermal stresses. The results obtained show that behavior of the temperature, displacement and stresses may change significantly by reason of influence of exponent heat flux and primary magnetic filed in homogeneous and nonhomogeneous cases. These results are specific for the example considered, but the example may have different trends because of the dependence of the results on the mechanical and thermal constants of the material. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
H. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309. A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elast. 2 (1972) 1–7. A.E. Green, P.M. Nagdhi, A re-examination of the basic postulate of thermo mechanics, Proc. Roy. Soc. London 432 (1991) 171–194. A.E. Green, P.M. Nagdhi, An undamped heat wave in elastic solid, J. Thermal Stresses 15 (1992) 253–264. A.E. Green, P.M. Nagdhi, Thermoelasticity without energy dissipation, J. Elast. 31 (1993) 189–208. E.S. Suhubi, Small torsional oscillations of a circular cylinder with finite electric conductivity in a constant axial magnetic field, Int. J. Eng. Sci 2 (1964) 441–459. J.C. Misra, S.C. Samanta, A.K. Chakrabarty, S.C. Misra, Magnetothermoelastic interaction in an infinite elastic continuum with a cylindrical hole subjected to ramp-type heating, Int. J. Eng. Sci. 29 (1991) 1505–1514. J.C. Misra, S.C. Samanta, A.K. Chakrabarty, Magneto-thermoelastic interaction in an aeolotropic solid cylinder subjected to a ramptype heating, Int. J. Eng. Sci. 29 (1991) 1065–1075. R.S. Dhaliwal, H.S. Saxena, G.J. Rokne, Generalized magnetothermoelastic waves in an infinite elastic solid with a cylindrical cavity, J. Thermal Stresses 14 (1991) 353–369. S.B. Mukhopadhyay, S.K. Roychoudhurj, Magneto-thermo-elastic interactions in an infinite isotropic elastic cylinder subjected to a periodic loading, Int. J. Eng. Sci. 35 (1997) 437–444. D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51 (1998) 705–729. Santwana Mukhopadhyay, Thermoelasic interactions without energy dissipation in an unbounded medium with a spherical cavity due to a thermal shock at the boundary, J. Thermal Stresses 25 (2002) 877–887. A.M. Abd-Alla, A.N. Abd-Alla, N.A. Zeidan, Thermal stresses in a non-homogeneous orthotropic elastic multilayered cylinder, J. Thermal Stresses 23 (2000) 413–428. A.M. El-Naggar, A.M. Abd-Alla, M.A. Fahmy, S.M. Ahmed, Thermal stresses in a rotating non-homgeneous orthotropic hollow cylinder, Heat Mass Transfer 39 (2002) 41–46. A.M. Abd-Alla, A.M. El-Naggar, M.A. Fahmy, Magneto-thermoelastic problem in non-homogeneous isotropic cylinder, Heat Mass Transfer 39 (2003) 625–629. S.B. Mukhopadhyay, Thermoelastic interactions without energy dissipation in an unbounded body with a spherical cavity subjected to harmonically varying temperature, Mech. Res. Commun. 31 (2004) 81–89.
A numerical solution of magneto-thermoelastic problem in non-homogeneous isotropic cylinder by the finite-difference method M.R. Abd-El-Salam a, A.M. Abd-Alla b, Hany A. Hosham
c,*
a
Department of Mathematics, Faculty of Science, South-Valley University, Sohag, Egypt Department of Mathematics, Faculty of Science, Taif University, Kingdom of Saudi Arabia c Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
b
Received 1 December 2005; received in revised form 1 April 2006; accepted 11 May 2006 Available online 7 August 2006
Abstract This paper presents the development of the magneto-thermoelastic problem in non-homogeneous isotropic cylinder in a primary magnetic field when the curved surface of the cylinder subject to certain boundary conditions. The governing coupled linear partial differential equations in the hyperbolic-type have been solved numerically using the finite-difference method. Graphical results for the temperature, displacement and components of stresses are illustrated and discussed for copper-like material. The results indicate that the effects of inhomogeneity and magnetic field are very pronounced. Some more interesting particular cases have also been discussed. 2006 Elsevier Inc. All rights reserved. Keywords: Magneto-thermoelastic; Coupled problems; Finite-difference method
1. Introduction The generalized theories of thermoelasticity, which admit the finite speed of thermal signal, have been the center of interest of active research during last three decades. The theory of generalized thermoelasticity was proposed by Lord and Shulman [1] and Green and Lindsay [2] hereafter called LS and GL theories, respectively. These theories have been developed by introducing one or two relaxation time in the thermoelastic process, with an aim to eliminate the paradox of infinite speed for the propagation of thermal signals. The two theories are structurally different form one another, and one cannot be obtained as a particular case of the other. Also, these theories involved a hyperbolic-type heat equation. These generalized theories were motivated by experiments exhibiting the actual occurrence of wave type heat transport and were considered to be more realistic than the conventional theory in dealing with practical problems involving very large heat fluxes and/or small time intervals, like those occurring in laser units and energy channels. *
Corresponding author. E-mail address: [email protected] (H.A. Hosham).
0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.05.009
M.R. Abd-El-Salam et al. / Applied Mathematical Modelling 31 (2007) 1662–1670
1663
The theory of magneto-thermoelasticity is concerned with the interacting effects of the applied magnetic field on the elastic and thermoelastic deformations of a solid body. This theory has aroused much interest in many industrial applications, particularly in nuclear devices, where there exists a primary magnetic field; various investigations are to be carried out by considering the interactions among magnetic, thermal and strain fields. Analysis of such problems also influence various applications in biomedical engineering as well as in different geomagnetic studies. The development of the interactions of an electromagnetic field, the thermal field, and the elastic field is a available in many studies see Green and Naghdi [3–5]. Among the authors who considered the generalized magneto-thermoelasticity equations is Suhubi [6] who studied magneto-thermo-viscoelastic interactions in a body having cylindrical geometry. Misra et al. [7,8] studied magneto-thermoelastic interactions in an infinite elastic medium with a cylindrical hole as well as in an aeolotropic solid cylinder subjected to a ramp-type heating. Dhaliwal et al. [9] discussed generalized magneto-thermoelastic wave in an infinite elastic solid with a cylindrical cavity. Mukhopadhyay and Roychoudhurj [10] discussed magneto-thermoelastic interactions in an infinite isotropic elastic cylinder subjected to a periodic loading. A review literature of hyperbolic thermoelasticity can be found in Chandrasekharaiah [11]. Recently, Mukhopadhyay [12] solved the thermoelastic interactions problem without energy dissipation in an unbounded medium, Abd-Alla et al. [13] have investigated the thermal stress in an infinite circular cylinder of orthotropic material. El-Naggar et al. [14] studied the thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Abd-Alla et al. [15] investigated the magneto-thermoelastic problem in nonhomogeneous isotropic cylinder. In the present paper, we have considered an magneto-thermoelastic problem which involved a hyperbolictype heat equation. We have discussed the generation of stress, temperature and magnetic field in an infinite isotropic elastic cylinder placed in a constant primary magnetic field. The conservation equation of an magneto-thermoelastic are first deduce in the non-homogeneous form. The governing equations are transformed into a dimensionless form and their solutions are then obtained by an finite-difference method. The outline of the coupled finite-difference solutions procedure, we compute the values of displacement and temperature on the first line by using initial conditions. Also, we use the central, forward and backward finite-difference approximations of a second-order accurate to eliminating fictitious points at initial and boundary conditions. Using initial and first lines we can compute the solutions at next time to arrive the final time. Finite-difference models seem to enjoy significant popularity for the ease with which they can be implemented. Numerical results are presented for the variation of the temperature, displacement, and stresses with the time and through the thickness of the cylinder. The effect of inhomogeneity and the magnetic field are very pronounced. 2. Formulation of the problem Let us consider a cylindrical coordinate system (r, h, z), for the axially symmetric problem, we have uh = 0, ur ¼ ur ðr; z; ^tÞ and uz ¼ uz ðr; z; ^tÞ, where ^t is the time. Furthermore, if only axisymmetric plane strain problem is considered, we have uh = uz = 0 and u ¼ ur ðr; ^tÞ. The strain–displacement relations are err ¼
ou ; or
u ehh ¼ ; r
ezz ¼ ezr ¼ erh ¼ ehz ¼ 0;
ð1Þ
where ui and eij are the displacement component and strain components, respectively. The stress–strain relations are ou u ou þ rrr ¼ k þ 2l c Tb ðr; ^tÞ; or r or ou ou u þ rhh ¼ k ð2Þ þ 2l c Tb ðr; ^tÞ; or or r ou u þ ; srr ¼ l0 H 20 or r
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where rij is the stress components, k and l are Lame’s constants, c = (3k + 2l)at, at is the coefficient of linear is the magnetic permeability and H0 is the primary thermal expansion, Tb ðr; ^tÞ is the absolute temperature, l magnetic field. Equation of motion in a linear homogeneous and isotropic elastic solid in the absence of body force is given by orrr 1 osrr o2 u þ ðrrr rhh Þ þ ¼q 2; r or or o^t
ð3Þ
where q is the mass density. The heat conduction equation when absence heat source is given by Mukhopadhyay [16] ! 2b o2 Tb o ou u 1 o Tb o T þ c 2 þ cT 0 2 þ ¼k ; or2 r or o^t o^t or r
ð4Þ
where T0 is the reference temperature, c = qcv is the specific heat at constant volume, k* is a material constant. In this study, we assume the non-homogeneous property of the material is characterized by k ¼ Lr2m ;
l ¼ vr2m ;
l0 ¼ v0 r2m ;
q ¼ q0 r2m ;
ð5Þ
0
where L, v, v and q0 are known constants. Substituting from Eq. (5) into Eq. (2) we obtain the stress–displacement relations are ou u ou 2m 0b þ rrr ¼ r L þ 2v c T ; or r or ou ou u 2m 0b þ rhh ¼ r L þ 2v c T ; ð6Þ or or r ou u þ ; srr ¼ r2m v0 H 20 or r where c 0 = (3L + 2v)at. Substituting from Eq. (6) into Eqs. (3) and (4) we obtain the equations of motion and heat conduction, respectively; ! b o2 u o2 u 1 ou u 2m o T þ ½2mðL þ v0 H 20 Þ g 2 c0 Tb þ q0 2 ¼ g 2 þ gð1 þ 2mÞ ; ð7Þ or r or r r or o^t ! 2 2b o2 Tb ou u 1 o Tb o T 0 2m o þ þ ¼k ; ð8Þ c 2 þ c T 0r or2 r or o^t o^t2 or r where g ¼ L þ 2v þ v0 H 20 . Now we introduce the non-dimensional quantities: k þ 2l r u; R¼ ; cT 0 a a rrr rhh ¼ ; rHH ¼ ; cT 0 cT 0
U¼ rRR
t t ¼ t; a
T ¼
sRR ¼
Tb ; T0
srr ; cT 0
ð9Þ
where t is a standard speed. The non-dimensional forms of Eqs. (7) and (8) are therefore obtained as o2 U o2 U 1 oU U 2m oT 2 þ a T þ ¼ c a þ a ; ð10Þ 1 2 3 2 p ot2 R oR R oR oR2 R 2 2 o2 T 1 oT oU U 2m o 2 o T þ ¼ cT þ eðaRÞ ; ð11Þ ot2 ot2 oR R oR2 R oR 2mðLþv0 H 2 Þg
02
c T0 g 0 where c2p ¼ Lþ2v , a1 ¼ Lþ2v , a2 ¼ gð1þ2mÞ , a3 ¼ , c2T ¼ ctk 2 , and e ¼ ðLþ2vÞc , and the non-dimensional forms Lþ2v Lþ2v qt2 of the stress–strain relation can be written are as follows:
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oU U þ k1 T ; rRR ¼ R oR R oU U 2m þ T ; rHH ¼ R k1 oR R v0 H 20 oU U þ sRR ¼ ; L þ 2v oR R
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2m
ð12Þ
k . where k1 ¼ kþ2l From preceding description, the initial condition may be expressed as
U ðR; 0Þ ¼
oU ðR; 0Þ ¼ 0; ot
T ðR; 0Þ ¼
oT ðR; 0Þ ¼ 0; ot
a 6 R 6 M;
ð13Þ
where a and M are inner and outer radii of the cylinder. The boundary conditions may be expressed as rRR ¼ 0; rRR ¼ 0;
T ¼ eXt ; at R ¼ a; t > 0; T ¼ 0; at R ¼ M; t > 0;
ð14Þ
where X is exponent of the decayed heat flux.
a
b
Fig. 1. Temperature distributions with m = 0.5 and H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5, and various values of X.
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3. Numerical solution of the problem The computational domain {(R, t) : R 2 [a, M], t 2 [0, 1)} is discretized into cells described by the node set (Ri, tn). The uniform mesh defined by Ri = a + ih; i = 0, 1, . . . , N; tn = nDt; n = 0, 1, . . . , K; h ¼ Ma is a mesh N width, Dt = tmax/K is the time step and tmax is the final time. Here, naturally, N and K are positive integers. In order to construct a numerical solution for the main problem (10)–(14) we introduce the explicit finitedifference scheme. Then the finite-difference equations corresponding to Eqs. (10) and (11) are as follows: a2 h n nþ1 n n1 U i ¼ 2U i U i þ q1 a1 ðU niþ1 2U ni þ U ni1 Þ þ ðU U ni1 Þ 2Ri iþ1 a3 h2 2mh2 n h n þ 2 U ni T i þ ðT iþ1 T ni1 Þ ; ð15Þ 2 Ri Ri h nþ1 n n1 n n n n n T i ¼ 2T i T i þ q2 ðT iþ1 2T i þ T i1 Þ þ ðT T i1 Þ 2Ri iþ1 where q1 ¼
eR2m e nþ1 n1 n n1 i ðU iþ1 2U niþ1 þ U iþ1 U nþ1 ðU nþ1 2U ni þ U n1 Þ; i1 þ 2U i1 U i1 Þ i Ri i 2h
c2p ðDtÞ2 h2
and q2 ¼
ð16Þ
c2T ðDtÞ2 . h2
a
b
Fig. 2. Radial displacement distributions with H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5, and various values of X.
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In order to start computation we require data on the two line t = 0, and t = Dt. The initial conditions (13) gives the initial line and values on the line t = Dt. Substituting the central difference approximation for the oU 0 oT 0 U 1 U 1 T 1 T 1 and T 1 from Eqs. (15) and (16) for n = 0, derivative, i.e. ot i ¼ i 2k i and oti ¼ i 2k i and eliminating U 1 i i we get the formulas to give the values on the first level. q1 a2 h 0 1 0 a1 ðU 0iþ1 2U 0i þ U 0i1 Þ þ Ui ¼ Ui þ ðU U 0i1 Þ 2Ri iþ1 2 a3 h2 2mh2 0 h 0 þ 2 U 0i T i þ ðT iþ1 T 0i1 Þ ; ð17Þ 2 Ri Ri q h T 1i ¼ T 0i þ 2 ðT 0iþ1 2T 0i þ T 0i1 Þ þ ðT 0iþ1 T 0i1 Þ 2Ri 2
eR2m e i ðU 1iþ1 U 0iþ1 U 1i1 þ U 0i1 Þ ðU 1i U 0i Þ: Ri 2h
ð18Þ
The boundary conditions at R = a, in the finite-difference form is U n1 U n1 Un þ k1 0 T n0 ¼ 0; 2h R0
T n0 ¼ eXt :
a
b
Fig. 3. Radial stress distributions with H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5 and various values of X.
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At R = a, t > 0 (i.e., i = 0), Eq. (15) becomes n a2 h 2hk1 U n0 nþ1 n n1 n n 2hk1 U 0 n n U 0 ¼ 2U 0 U 0 þ q1 a1 2U 1 2U 0 2T 0 þ þ 2T 0 2R0 R0 R0 a3 h2 2mh2 n h þ 2 U n0 T þ ð3T n0 þ 4T n1 T n2 Þ : R0 0 2 R0
ð19Þ
Also, the boundary conditions at R = M, in the finite-difference form is U nN þ1 U nN 1 Un þ k1 N T nN ¼ 0; 2h RN
T nN ¼ 0:
At R = M, t > 0 (i.e., i = N), Eq. (15) becomes 2hk1 U nN a2 h 2hk1 U nN n n n n U Nnþ1 ¼ 2U nN U n1 þ q a 2U 2U 2T þ 2T þ 1 1 N N 1 N N N 2RN RN R0 a3 h2 2mh2 n h þ 2 U nN T þ ð3T nN 4T nN 1 þ T nN 2 Þ : RN N 2 RN
ð20Þ
a
b
Fig. 4. Hoop stress distributions with H0 = 3.0 · 105. (a) X = 1.0, and various values of t. (b) t = 0.5 and various values of X.
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The local truncation error is a second-order accurate in time and space O((Dt)2 + h2), and it’s tends to zero as Dt and h tend to zero. Hence the present method is compatible. 4. Numerical results and discussion In order to illustrate the problem, the copper material was chosen for purposes of numerical evaluations. The physical data which given as e = 0.0168, Ł = 1.387 · 1012 dyne/cm2, v = 0.448 · 1012 dyne/cm2, cp = 1.0, cT = 0.578 and v 0 = 1.0 Gauss/Oersted. Results are presented for cylinder with a = 1.0, M = 3.0. Fig. 1 shows the temperature variation in the non-homogeneous case when the primary magnetic field is constant H0 = 3.0 · 105, for various values of non-dimensional time, t (Fig. 1a) or various values of exponent decayed heat flux, X (Fig. 1b). It is noticed that the temperature decreases with increasing of t and X, and the solutions satisfied the boundary conditions. Fig. 2 presents the radial displacement distributions along radial direction R with the various values of the t and X. The values of displacement are also seen to increase with increasing of time and arrived maximum value at t = 0.5, but the displacement decrease with increasing exponent of the decayed heat flux. Also, we can find the difference between the non-homogeneous and homogeneous problems, beside, this difference increases with increase of t and decrease of X. Figs. 3 and 4 show the influence of the non-homogeneous of the material constants on the radial and hoop stresses with the same values of magnetic field H0 and the parameters as those in Fig. 2. Also, they show the difference among two cases namely non-homogeneous and homogeneous bodies.
a
b
c
Fig. 5. Radial displacement, radial stress and hoop stress distributions, respectively for various values of H0 with t = 0.5, and X = 1.0.
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The coupling effect resists the temperature rise, and the resisting effect increases with increasing time. The physical meaning of this phenomenon is that the coupling effect acts like a damper inside the material, reduces the rate of heat propagation, and thus slows down the temperature rise. The influence of the primary magnetic field and inhomogeneity on the radial displacement, radial stress and hoop stress are show in Fig. 5. we can see that, the solutions satisfied the boundary conditions, and the difference between homogeneous and non-homogeneous cases can be shown. The variation of temperature, displacement, and stresses are due to the effect of inertia and magnetic field. Also, the influence of the nonhomogeneous on displacement and stresses is very pronounced. 5. Conclusion In this paper, a second-order explicit finite-difference scheme has been described. The scheme solves the coupled hyperbolic equations on a uniform grid, and is quite efficient for computation thermal stresses. The results obtained show that behavior of the temperature, displacement and stresses may change significantly by reason of influence of exponent heat flux and primary magnetic filed in homogeneous and nonhomogeneous cases. These results are specific for the example considered, but the example may have different trends because of the dependence of the results on the mechanical and thermal constants of the material. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
H. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309. A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elast. 2 (1972) 1–7. A.E. Green, P.M. Nagdhi, A re-examination of the basic postulate of thermo mechanics, Proc. Roy. Soc. London 432 (1991) 171–194. A.E. Green, P.M. Nagdhi, An undamped heat wave in elastic solid, J. Thermal Stresses 15 (1992) 253–264. A.E. Green, P.M. Nagdhi, Thermoelasticity without energy dissipation, J. Elast. 31 (1993) 189–208. E.S. Suhubi, Small torsional oscillations of a circular cylinder with finite electric conductivity in a constant axial magnetic field, Int. J. Eng. Sci 2 (1964) 441–459. J.C. Misra, S.C. Samanta, A.K. Chakrabarty, S.C. Misra, Magnetothermoelastic interaction in an infinite elastic continuum with a cylindrical hole subjected to ramp-type heating, Int. J. Eng. Sci. 29 (1991) 1505–1514. J.C. Misra, S.C. Samanta, A.K. Chakrabarty, Magneto-thermoelastic interaction in an aeolotropic solid cylinder subjected to a ramptype heating, Int. J. Eng. Sci. 29 (1991) 1065–1075. R.S. Dhaliwal, H.S. Saxena, G.J. Rokne, Generalized magnetothermoelastic waves in an infinite elastic solid with a cylindrical cavity, J. Thermal Stresses 14 (1991) 353–369. S.B. Mukhopadhyay, S.K. Roychoudhurj, Magneto-thermo-elastic interactions in an infinite isotropic elastic cylinder subjected to a periodic loading, Int. J. Eng. Sci. 35 (1997) 437–444. D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51 (1998) 705–729. Santwana Mukhopadhyay, Thermoelasic interactions without energy dissipation in an unbounded medium with a spherical cavity due to a thermal shock at the boundary, J. Thermal Stresses 25 (2002) 877–887. A.M. Abd-Alla, A.N. Abd-Alla, N.A. Zeidan, Thermal stresses in a non-homogeneous orthotropic elastic multilayered cylinder, J. Thermal Stresses 23 (2000) 413–428. A.M. El-Naggar, A.M. Abd-Alla, M.A. Fahmy, S.M. Ahmed, Thermal stresses in a rotating non-homgeneous orthotropic hollow cylinder, Heat Mass Transfer 39 (2002) 41–46. A.M. Abd-Alla, A.M. El-Naggar, M.A. Fahmy, Magneto-thermoelastic problem in non-homogeneous isotropic cylinder, Heat Mass Transfer 39 (2003) 625–629. S.B. Mukhopadhyay, Thermoelastic interactions without energy dissipation in an unbounded body with a spherical cavity subjected to harmonically varying temperature, Mech. Res. Commun. 31 (2004) 81–89.