 World Journal of Mechanics, 2013, 3, 6-12 http://dx.doi.org/10.4236/wjm.2013.35A002 Published Online August 2013 (http://www.scirp.org/journal/wjm) Thermoelastic Problem of a Long Annular Multilayered Cylinder Yi Hsien Wu1*, Kuo-Chang Jane2 1Department of Information Management, Oriental Institute of Technology, Taipei, Chinese Taipei 2Department of Applied Mathematics, National Chung Hsing University, Taichung, Chinese Taipei Email: *yhwu@mail.oit.edu.tw Received May 2, 2013; revised June 2, 2013; accepted June 9, 2013 Copyright © 2013 Yi Hsien Wu, Kuo-Chang Jane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Thermoelastic transient response of multilayered annu lar cylinde rs of infinite leng ths subjected to known inner pressure and outer surfaces cooling are considered. A method based on the Laplace transformation and finite difference method has been developed to analyze the thermoelasticity problem. Using the Laplace transform with respect to time, the gen-eral solutions of the governing equations are obtained in transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. Solutions for the temperature and thermal stress distributions in a transient state were obtained. It was found that the temperature distribution, the displacement and the thermal stresses change slightly as time increases. Keywords: Thermoelastic; Multilayered Annular Cylinders; Laplace Transformation; Finite Difference Method 1. Introduction A thermal problem arises when the composed materials are generated by a sudden change in temperature. Shell structures are widely used in contemporary industries, so we must take care of the thermal problem. The shell structures may be affected due to the pressure change or the various temperature distributions. It is necessary to solve for temperature or pressure at first. The dynamic thermoelastic response of circular shell rapidly change of thermal environments is important for the design of many engineering structures. Due to the complexity of the governing equations and the mathe-matical difficulties associated with the solution, several simplifications have been used. For example, Sherief and Anwar  discussed the problem of an annular infinitely long elastic circular. They have neglected both the in ertia terms and the relaxation effects of the problem. Sherief and Anwar  considered the thermoelasticity problem of an infinitely long annular cylinder composed of two different materials with axial symmetry. The solution was obtained in the Laplace transform domain by using the potential functio n approach. The present work deals with the one-dimensional qua-sistatic coupled thermoelastic problems of an infinitely long annular multilayered cylinder composed of multi-layered different materials. The medium has a pressure at the inner layer, the temperature to be heated at the outer layer, without body forces and internal heat generation. Derivatives are approximated by central differences re-sulting in an algebraic representation of the partial dif-ferential equation. By taking the Laplace transform with respect to time, the general solutions in the transform domain are first obtained. The final solutions in the real domain can be obtained by inverting the Laplace trans-form. 2. Formulation This work deals with the one-dimensional, quasi-static coupled, thermoelastic problems of an infinitely long annular cylinder composed of multilayered laminated materials with axial symmetry under the following as-sumptions: 1) Materials of each layer are assumed to be non-homogeneous; 2) Deformation and strain satisfy the Hooke’s law and small strain theory; 3) The composite cylinder is constructed of multilayered laminates bonded together p erfectly; 4) Th e medium is initiall y undisturb ed, and without body forces and internal heat sources; 5) The medium is applied by a force, which is the function of time; 6) The temperature at inner layer and outer layer are the functions of time. We now consider an infinitely long annular cylinder *Corresponding a uthor. Copyright © 2013 SciRes. WJM Y. H. WU, K.-C. JANE 7made of multiple layers of different materials. The inner and outer radii of the cylinder are denoted by i and o, respectively. The multilayered composite is assumed to be heated suddenly at the inner and outer surface under temperatures r r1f and 2f respectively. The transient heat conduction equation for the layer in dimensional form can be written as (see Equation (1) below) thiwhere 0 1rrrrrEr and 1rrrrrEr in which is the radial component of displacement, is radius, and UrvC are specific heat and density of material, r and r are the Poisson’s ratio , rkk are radial, circumferential thermal conductivity, r,  are radial and circumferential thermal expansion coefficient, r, E E are radial and circumferential Young’s modulus, , 0 are the temperature, refer-ence temperature, and  is time, respectively. If the body forces are absent, the equation of equilib-rium for a cylinder along the radial direction can be writ-ten as 222111rrrrrrrr rrrEEUUUEErrrrEErr     (2) The stress-displacement relations are 0rrri riiiEEU Urr    (3) 0rriiiiEEUUrr    (4) where , r,  are Lame’s constant, radial and circumferential stresses respectively. Let the boundary conditions of multilayered cylinder be at 0t 0U at 1rR 10,ectrrt P 10at orR ,0rrt 202 1ectout rf  where 1f, 2f, 0, rP are inner and outer surround-ing temperatures, initial inner pressure, the initial tem-perature at the outer layer respectively. The non-dimensional variables are defined as fol-lows: 00T0 22 221cos sincos sinrrivvikk kkaCC 22 221sincoscos sinrrivvikk kkbCC 22 22sincoscos sinrrivviiwCC  22211cos sinrvkktRC 1rrR 221cos sinrviuU RC 1irr rieEE  irifEE 220cos sinririvigCr 220cos sinrirrvr iiEhCE   rr  2211cos sinrririviEQC0 2221cos sinrririviEQC 0 31irirQ 2211cos sinrr riiviERC0 2221cos sinrriiviERC0 1f  2222 222202201cos sinsincoscossin1sin cosrr vrrUkk kkCrr rrUr      (1)Copyright © 2013 SciRes. WJM Y. H. WU, K.-C. JANE 8 31iiR 10riri r 10ii where , , Ttr, , riu, iare non-dimensional temperature, time, radius, displacement, radial stress and circumferential stress for the layer respectively. ithSubstituting the nondimensional quantities into the governing Equations (1)-(4), the transient heat conduc-tion equation and stress-displacement relations have the following nondimensional form: 22iibuaTrrtr trtr      222iiiieuuu TTfghrrrrrr   (6) 123ri iiiuuQQQrrT (7) 123iii iuuRRRrrT (8) iwu (5) 3. Computational Procedures Applying central difference in Equations (5)-(8), we ar-rive at the following discretized equations: 1111 122122jjjjjjjjjiiijj jjjuuTTTTTTu ttwabrr trtrr       1 (9) 1111 112221122jjjjjj jiijijijjjjjuuuuuTTefugrr rrr  jThr (10) 11122jjjriiii jjjuu uQQrr3QT (11) 11122jj jiii ijjuu uRRrr3jRT (12) where 11jrN  and 1, 2,,jN. The Laplace transform of a function are de-fined by t 0edstsLt t t Take the Laplace transform for Equations (9)-(12), we obtain the following equations:  1111,,21,1 1,121()212jjjjjiiijin jjin jjj jjjinj jinjjTTTTTwabTsTusurr rrusuusur  (13) 1111 112221122jjjjjj jjiijiijjjjjuuuuuTT Tefugrr rrr  jhr (14) Let the surface of the cylindrical inner surface be stress free and subject to a time-dependent temperature. After taking Laplace transformation, the boundary condi-tions in transformed domain become 010 11,rrPrssc 11Tfs0 at ; 1rr,rrs0 1211rTssc  at . outrrand the interface conditions are as follows: 1,iiursu rs, 1irr 1,ri rirs rs, 1irr 1,iiqrsqrs, 1irr 1,iiTrs T rs, 1irr where 1, 2,,1im. Substituting the boundary conditions and the interface conditions into Equations (13), (14), we obtain the fol-Copyright © 2013 SciRes. WJM Y. H. WU, K.-C. JANE 9lowing equation in matrix form (see Equations (15) be-low) where 131 11211 121QaBQr  22ijiaBr  131 21 1111111 111QQwEQrQr  212iijjiiabCrrr 1110NN NFD X YZG 12jiDr 1321112mmmNmkmQabAQrr  mr ijjwEr 132121mmNmmQaBQr  12jiFr 1321111mmmNmkmkr131 0111111QPXQQ 0jY 132 212112mmNrkmmQabYQrrr QwQEQrQ   0jX 0Z j132 212112mmNrkmmQabZQrrr 0G j131 011112111 1111112QPbaGTQrrQr   where denotes the last layer, the last point, and denotes it layer for mkih2,3, ,1jN, (see Equation (16) below) where: 2ijigHr ijjhIr 2ijigJr  2112ijjiieKrrr 222ijjifLrr  2112ijjiieMrrr 2,3, ,1jN 11 11122 222212111 1111122111NNN NNNNN NNNBC TXYAB CTXYsI scsc sABC TXYAB TXYEFDE Fs  121NNZZZZ         1122211111NNNN NNNN NuGuGDEFuGDEu G  (15)111 1112222 222211 11 11110000NN NN NNNNNNN NNNIJLM uTHIJKL MuTHIJKLM uTHIK LuT            (16)Copyright © 2013 SciRes. WJM Y. H. WU, K.-C. JANE 10 Equations (15) and (16) can be rewritten in the following matrix forms    12111jjjjjjMsITsN uXscYZGsc s(17) 0jjRT Qu (18) where the matrix M, N, R and Q are the corresponding matrix in Equations (15) and (16). Substi-tuting Equation (17) into (18), we have    12111jjjjjAsI TBscCDFsc s (19) where   111 1ANQRNM  111 1jjBNQRNX   111 1jjCNQRNY  111 1jjDNQRNZ   111 1jjFNQRNG  Since the matrix NNA is a nonsingular real matrix, the matrix A possesses a set of linearly independent eigenvectors, hence the matrix NA is di-agonalizable. There exist a nonsingular transition matrix P such that , that is, the ma-trices  P gA1PAdia A and diag A are similar, where the ma- trix diag1, 2jA,N is a diagonal matrix with elements j, where ,j is the eigenvalue of ma-trix A. The equation ca n be o bt ai n ed as 11111121111jjjjPAPsIPT PBscPCPD PFsc sj (20) Equation (20) can be rewritten as    12111jjjjjdiagAs ITBscCDFsc s (21) where 1jjTPT, 1jjBPB 1jjCPC, 1jjDPD and 1jjFPF From Equation (21), the following solutions can be obtained immediately. 12jj jjii iFBC DTjisscsscs ss    (22) By applying the inverse Laplace transforms to Equa-tion (22), we get the solution jT. The eigenvalue, ei-genvector and inverse Laplace transform of matrix can be solved by applying the IMSL MATH/LIBRARY subroutines. ][AAfter we have obtained jT, then we can use Equa-tions (23) and (24) to obtain the solutions jT and ju jjTPT (23) 1jjuQRT (24) Substituting jT and ju into Equations (11) and (12), we obtain the radial and circumferential stresses. 4. Numerical Results and Discussions In this section, we present some numerical results of the temperature distribution in a long multilayered co mposite hollow cylinder, and displacement and thermal stresses under temperature changes. The inner and outer radii of the cylinder are assumed to be 1.0 and 4.5 respectively. For an infinitely long an-nular multilayered cylinder, the geometry and material quantities of the cylinder (in the case of three layers, layer 1:E = 58E6, k = 22,  = 0.2,  = 2.8E – 6,  = 0.095, Cv = 0.31 and layer 2 :E = 30E6, k = 21,  = 0.35,  = 2.3E – 6,  = 0.053, Cv = 0.25 and layer 3 : E = 22E6, k = 17,  = 0.2,  = 2.8E – 6,  = 0.09, Cv = 0.17 ; in the case of five layers, layer 1 : E = 58E6, k = 22,  = 0.2,  = 2.8E – 6,  = 0.095, Cv = 0.31 and layer 2 : E = 30E6, k = 21,  = 0.35,  = 2.3E – 6,  = 0.053, Cv = 0.25 and layer 3:E = 22E6, k = 17,  = 0.2,  = 2.8E – 6,  = 0.09, Cv = 0.17 and layer 4:E = 30E6, k = 21,  = 0.35,  = 2.3E – 6,  = 0.053, Cv = 0.25 and layer 5 : E = 22E6, k = 17,  = 0.2,  = 2.8E – 6,  = 0.09, Cv = 0.17). Each layer is as-sumed to have a different thickness (in the case of three layers, r1 = 1.5, r2 = 0.5 and r3 = 1.5; in the case of five layers, r1 = 1.0, r2 = 0.5, r3 = 1.0, r4 = 0.5 and r5 = 0.5). The pressure of the inner surface is assumed to be P0 = 1.5E6. The constant coefficient c1 = c2 = 1.0. The tem-perature at inner surface is assumed to be 300, at outer Copyright © 2013 SciRes. WJM Y. H. WU, K.-C. JANE 11surface which is a function of time is assumed to be 0 to 100. Figures 1-4 show some numerical results of three and five layered cylinders at time step t = 0.5, 1, 2, 5 and 10. Figures 1 and 2 show the temperature distributions along radial direction for 3 and 5 layers case. Because of the difference in thermal conductivity and the effect of the outer layer is to be heated. As time is small, say t = 0.5, the outer layer temperature which is to be heated is not so more, so the distribution decreasing at first and then increasing. Figures 3 and 4 show the dis-placement along the radial direction. The maximum dis-placement occurred at the interface of first and second layers. Figures 5 and 6 show the radial stress distribution r along the radial direction. Figures 7 and 8 show the circumferential stress  along the circumferential di-rection. Figure 1. Temperature distribution along radial direction for 3 layers case. Figure 2. Temperature distribution along radial direction for 5 layers case. Figure 3. Radial displacement distribution along radial direction for 3 layers case. Figure 4. Radial displacement distribution along radial direction for 5 layers case. Figure 5. Radial stress distribution along radial direction for 3 layers case. Copyright © 2013 SciRes. WJM Y. H. WU, K.-C. JANE Copyright © 2013 SciRes. WJM 12 Figure 8. Circumferential stress distribution along radial direction for 5 layers case. Figure 6. Radial stress distribution along radial direction for 5 layers case. distributions have been obtained, all of which can be used to design useful structures or machines for engi-neering applications. There is no limit to the number of annular layers in a cylinder. Exemplifying numerical results from three- and five- layered cylinders at different time steps have been presented. The discontinuity in cir- cumferential stress at each interface was found. It was found that the temperature distribution, the displacement and the thermal stresses vary slightly as the time in- creases. REFERENCES  H. H. Sherief and M. N. Anwar, “Problem in Gene ralized Thermoelasticity,” Journal of Thermal Stresses, Vol. 9, No. 2, 1986, pp. 165-181. doi:10.1080/01495738608961895  H. H. Sherief and M. N. Anwar, “A Problem in General-ized Thermoelasticity for an Infinitely Long Annular Cylinder Composed of Two Different Materials,” Acta Mechanica, Vol. 80, No. 1-2, 1989, pp. 137-149. doi:10.1007/BF01178185 Figure 7. Circumferential stress distribution along radial direction for 3 layers case. A method based on the finite difference and Laplace transformation has been developed to obtain numerical results. The temperature, displacement and thermal stress