An Exact Solution for Classic

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M. JabbariAssistant Professor

Postgraduate School,Islamic Azad University, South Tehran Branch,

Tehran, Iran

H. DehbaniPostgraduate School,

Sama Technical and Vocational Training School,Islamic Azad University, Varamin Branch,

Varamin, Iran

M. R. EslamiProfessor

Fellow ASMEDepartment of Mechanical Engineering,

Amirkabir University of Technology,Tehran, Iran

An Exact Solution for ClassicCoupled Thermoelasticity inCylindrical CoordinatesIn this paper, the classic coupled thermoelasticity model of hollow and solid cylindersunder radial-symmetric loading condition �r , t� is considered. A full analytical method isused, and an exact unique solution of the classic coupled equations is presented. Thethermal and mechanical boundary conditions, the body force, and the heat source areconsidered in the most general forms, where no limiting assumption is used.�DOI: 10.1115/1.4003459�

Keywords: coupled thermoelasticity, hollow cylinder, exact solution

IntroductionThe coupled thermoelasticity of structural problems is fre-

uently referred to in literature, where the assumption is used inhe advanced engineering design problems for the structures underhermal shock loads. A thick circular cylindrical pressure vessel,s a structural member, may be subjected to mechanical and ther-al shock loads. A pressure vessel under thermal shock at refin-

ries and power plants, when the period of thermal shock is of theame order of magnitude as the period of lowest natural frequencyf the vessel, may experience the stress wave fronts and thushould be analyzed through the coupled form of the energy andhermoelasticity equations. In this case, thermal stress and tem-erature wave fronts are produced and propagated through thetructure. Engineering codes consider such a case as one of theoading conditions and put a limit on the magnitude of the result-ng stresses by defining a proper allowable stress value �1�.

The literature on the subject of coupled thermoelasticity for aounded geometry with the given boundary conditions is ex-remely limited. The coupled thermoelasticity of an infinite spacend a half-space is treated in literature, and exact solutions arevailable. For these cases, a few number of papers that present thelosed-form or analytical solution for the coupled thermoelasticityroblems are available. Hetnarski �2� found the solution ofoupled thermoelasticity in the form of a series function. Hetnar-ki and Ignaczak �3� presented a study of the one dimensionalhermoelastic waves produced by an instantaneous plane source ofeat in homogeneous isotropic infinite and semi-infinite bodies ofhe Green–Lindsay type. The same authors presented an analysisf the laser-induced waves propagating in an absorbing ther-oelastic semispace of the Green–Lindsay type �4�. Georgiadis

nd Lykotrafitis �5� obtained a three dimensional transient ther-oelastic solution for the Rayleigh-type disturbances propagating

n the surface of a half-space. Wagner �6� presented the funda-ental matrix of a system of partial differential operators that

overns the diffusion of heat and the strains in an elastic media.his method can be used to predict the temperature distributionnd the strains by an instantaneous heat point source or by auddenly applied delta force.

The mathematical treatment of such thermoelasticity problemsy the analytical methods is rather complicated, and the numerical

Contributed by the Pressure Vessel and Piping Division of ASME for publicationn the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received February 20,010; final manuscript received January 3, 2011; published online July 14, 2011.
ssoc. Editor: Somnath Chattopdhyay.
ournal of Pressure Vessel Technology Copyright © 20

nloaded 29 Nov 2011 to 115.114.107.36. Redistribution subject to ASM

methods of solution are employed to analyze such problems.Milne et al. studied spherical elastic-plastic wave solutions �7�.These authors applied two closed-form radially symmetric elastic-plastic wave solutions corresponding to the spherical cavity load-ings and compared the results with the numerical solutions con-structed by a finite difference code. Berezovski et al. presented thenumerical simulation of nonlinear elastic wave propagation inpiecewise homogeneous media �8�. Also, Berezovski et al. pre-sented the two dimensional propagation of stress waves in func-tionally graded materials �FGMs� using a numerical method andby means of the composite wave-propagation algorithm �9�. Theyconsidered two distinct models of FGMs: first, a multilayeredmetal-ceramic composite with averaged properties within the lay-ers and, second, the randomly embedded ceramic particles in ametal matrix with prescribed volume fractions. Their analysisshowed significant differences in the stress wave characteristicsfor the two distinct models, which can be used for optimizing theresponse of such structures to impact loading. Berezovski andMaugin simulated the wave front propagation in thermoelasticmaterials with phase transformation by means of the finite-volumenumerical method �10�. These authors also simulated the ther-moelastic wave propagation by means of a composite wave-propagation algorithm �11�. Engelbrecht et al. studied the nonlin-ear effects on the deformation waves in solids and dispersion �12�.In the mentioned work, the existence of solitary waves, the emer-gence of solitary wave trains, and the waves in piecewise nonlin-ear laminated materials are briefly discussed. Angel and Achen-bach studied the effect of oblique incidence with an arbitraryangle on the reflection and transmission of elastic waves by aperiodic array of cracks, and they studied the interaction of elasticwaves with a planar array of periodically spaced cracks of equallengths �13�. Mendelsohn et al. presented the two dimensionalscattering of incident surface waves and incident body waves by asurface-breaking crack �14�. Dempsey et al. computed the mode-III stress-intensity factor at the tip of the kinked crack for anglesof incidence varying from normal to grazing incidence and forarbitrary subsonic crack tip speeds �15�. Achenbach derived ex-plicit solutions for carrier waves supporting the surface waves andplate waves �16�. He reconsidered time-harmonic surface wavesin a half-space and plate waves in a layer of a homogeneoustransversely isotropic linearly elastic solid, where the x1x2-plane isthe plane of transverse isotropy. Achenbach and Li studied the twodimensional propagation of horizontally polarized transversewaves in a solid with a periodic distribution of cracks �17�. Theyused the theory of Floquet or Bloch waves, together with an ap-
propriate Green’s function and the condition of vanishing traction
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n the crack faces, which leads to a system of singular integralquations, which provides the basis for the derivation of an exactispersion equation. Roberts et al. investigated analytically andxperimentally the reflection of elastic waves from a traction-freeolid-air boundary of periodic saw-tooth profile �18�. Brind et al.tudied the high-frequency scattering of elastic waves from theylindrical cavities �19�. In their paper, the scattering of time-armonic plane longitudinal elastic waves by smooth convex cy-indrical cavities is investigated, and the exact solution for a circles evaluated for wavelengths of the same order as the radius. Auldtudied the acoustic fields and wave propagation in both isotropicnd anisotropic solids �20�. He discussed the reflection and refrac-ion at plane surfaces, composite media, waveguides, and resona-ors. Achenbach studied the wave propagation in elastic solids andresented an exposition of the basic concepts of mechanical waveropagation within a one dimensional setting. He studied the for-al aspects of elastodynamic theory in three dimensions and dis-

ussed the typical wave-propagation phenomena, such as radia-ion, reflection, refraction, propagation in waveguides, andiffraction �21�.

In problems of coupled thermoelasticity, the majority of papersocus on the numerical methods of solution. Bagri and Eslami22� studied the generalized coupled thermoelasticity of a func-ionally graded annular disk based on the Lord–Shulman model,here the Laplace transform is used to transform the governing

quations into the Laplace domain. The Galerkin finite elementethod is employed to solve the system of ordinary differential

quations in the space domain, where the actual physical quanti-ies in the time domain are obtained using the numerical inversionf the Laplace transform. Lee and Yang �23� investigated an in-erse problem of the coupled thermoelasticity of an infinite cylin-er and estimated the time-varying heat flux of the cylinder at theuter boundary by the time history of the measured temperaturen an interior point or distributed on the surface. Yang et al. �24�resented a technique to solve the inverse boundary value prob-ems of coupled thermoelasticity in an infinitely long annular cyl-nder and computed the boundary time-varying heat flux by know-ng the strain history at any point of the cylinder. Subsequently,he distributions of temperature and thermal stresses in the cylin-er at various times are determined. Eraslan and Orean �25� ob-ained the transient solution of the thermostatic-plastic deforma-ion of internal heat-generating tubes by the thermomechanicaloupling effect and temperature-dependent physical material pa-ameters and used the partial differential solver PDECOL for thisurpose. PDECOL is based on the method of lines and uses a finitelement collocation procedure for the discretization of the spatialariable. Yang and Chu �26� analyzed the transient coupled ther-oelasticity of an annular fin and neglected the effect of the in-

rtia term in the equation of motion. They considered the me-hanical coupling effect in heat conduction equation. Using theaplace transform with respect to time, the governing equationsecame decoupled, and the approximate method �Fourier seriesechnique� was used to achieve the inversion to the real time do-

ain. Bahtui and Eslami �27� studied the coupled thermoelasticesponse of a functionally graded circular cylindrical shell andsed a Galerkin finite element formulation in the space domainnd the Laplace transform in the time domain. Bakhshi et al. �28�tudied the coupled thermoelasticity of a functionally graded disknd used the Laplace transform and the Galerkin finite elementethod to solve the governing equations. Hosseni-Tehrani andslami �29� used the boundary element method to develop theoupled thermoelasticity formulation of the thermal and mechani-al shock problems in a two dimensional finite domain. The for-ulation is based on the Laplace transfer technique in the time

omain, where the solution is obtained using an appropriate nu-erical inversion technique. Tanigawa and Takeuti �30� developednew technique for the coupled thermal stress problem of a hol-

ow sphere under partial heating. The solution is obtained by a
imultaneous determination of the stresses and temperature distri-
51204-2 / Vol. 133, OCTOBER 2011


bution by introducing a new harmonic function. They used theLaplace transform and Laplace inversion formulas with the theo-rem of residue. Bagri and Eslami �31� presented a solution for theone dimensional generalized thermoelasticity of a disk. They em-ployed the Laplace transform and the Galerkin finite elementmethod to solve the governing equations. Bagri et al. �32� pro-posed a unified formulation of the generalized coupled ther-moelasticity and applied it to a layer of isotropic material, wherethe governing equations are solved by the Laplace transform andthe numerical inverse of the Laplace transform. Cannarozzi andUbertini �33� presented a mixed variational method for the linearcoupled thermoelasticity, where a finite element model for thesemidiscrete analysis is developed.

In this work, a full analytical method is proposed to obtain theresponse of the governing equations of the classical coupled ther-moelasticity in cylindrical coordinates, where an exact solution ispresented. The method of solution is based on the Fourier expan-sion and eigenfunction methods, which is a traditional and routinemethod in solving the partial differential equations. Since the co-efficients of equations are not functions of the time variable �t�, anexponential form is considered for the general solution. For theparticular solution, that is, the response to mechanical and thermalshocks, the eigenfunction method and Laplace transformation areused. This work is the extension of the previous paper that pre-sented an exact solution in the spherical coordinates �34�.

2 Governing EquationsA hollow cylinder with inner and outer radii ri and ro made of

isotropic material subjected to the radial-symmetric mechanicaland thermal shock loads is considered. The classical theory ofcoupled thermoelasticity for the wave propagation is considered.If u is the displacement component in the radial direction, thestrain-displacement relations for the radial-symmetric loadingcondition are

�rr = u,r

�1�

�� =u

r

where �,� denotes the partial derivative. The stress-strain relationsfor the plane strain condition are

�rr =E

�1 + ��1 − 2��1 − ��rr + �� −

E�

�1 − 2��T�r,t�

�2�

�� =E

�1 + ��1 − 2��rr + �1 − �� −

E�

�1 − 2��T�r,t�

The equation of motion in the radial direct is

�rr,r +1

r��rr − �� + F�r,t� = �u �3�

Using relations �1�–�3�, the Navier equation in terms of the dis-placement component is

u,rr +1

ru,r −

1

r2u − ��1 + ��1 − ��

T,r −�

E

�1 + ��1 − 2��1 − ��

u = − F�r,t�

�4�The coupled heat conduction equation for the radial-symmetricloading condition is �34,35�

T,rr +1

rT,r −

�c

kT −

E�T�

k�1 − 2��u,r +

1

ru� = − Q�r,t� �5�

where �, �, E, k, c, and T� denote the linear thermal coefficient,density, modulus of elasticity, thermal conduction coefficient, spe-cific heat, and initial reference temperature, respectively. Here,
F�r , t� and Q�r , t� are the body force and heat generation source,
Transactions of the ASME


r�

witcppg

3

dr

Cdv

S�

EtrA

Sra

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espectively. The mechanical and thermal boundary conditions are34�

C11u�ri,t� + C12u,r�ri,t� + C13T�ri,t� = f1�t�

C21u�ro,t� + C22u,r�ro,t� + C23T�ro,t� = f2�t��6�

C31T�ri,t� + C32T,r�ri,t� = f3�t�

C41T�ro,t� + C42T,r�ro,t� = f4�t�

here Cij are the mechanical and thermal coefficients; by assign-ng different values for them, different types of mechanical andhermal boundary conditions may be obtained. These boundaryonditions include the displacement, strain, stress, specified tem-erature, convection, and heat flux, and they are presented in Ap-endixes B and C. The initial boundary conditions are assumed ineneral form,

u�r,0� = f5�r�

u,t�r,0� = f6�r� �7�

T�r,0� = f7�r�

SolutionEquations �4� and �5� are a system of nonhomogeneous partial

ifferential equations with nonconstant coefficients �functions ofadial variable r� and with general and particular solutions.

3.1 General Solution With Homogeneous Boundaryonditions. Since the coefficients of these equations are indepen-ent of time variable �t�, the exponential function form of the timeariable may be assumed for the general solution as

u�r,t� = �U�r��e�t

�8�T�r,t� = ��r��e�t

ubstituting Eq. �8� into the homogeneous parts of Eqs. �4� and5� yields

U� +1

rU� −

1

r2U + d1�� + d2�2U = 0

�9�

�� +1

r�� + d3�� + d4��U� +

1

rU� = 0

quation �9� is a system of ordinary differential equations, wherehe prime symbol � �� shows differentiation with respect to theadial variable �r� and d1–d4 are constant parameters given inppendix A.The first solutions of U1 and �1 are considered as

U1�r� = A1J1��r��10�

�1�r� = B1J0��r�ubstituting Eq. �10� into Eq. �9� and using the formulas for de-ivatives of the Bessel function, such as dJ0��r� /dr=−�J1��r�nd dJn��r� /dr=�Jn−1��r�−n�Jn��r� /r�, yield

��− �2 + �2d2�A1 − d1�B1�J1��r� = 0�11�

��d4�A1 + �− �2 + �d3�B1�J0��r� = 0

quation �11� shows that U1 and �1 can be the solutions of Eq. �9�f and only if

�− �2 + �2d2� − d1�

�d4� �− �2 + �d3� �A1

B1� = �0

0� �12�

he nontrivial solution of Eq. �12� is obtained by setting the de-
erminant of this equation equal to zero as
ournal of Pressure Vessel Technology


�− �2 + �2d2��− �2 + �d3� + d1d4�2� = 0 �13�

Equation �13� is the first characteristic equation. Thus, it is con-cluded that the first solutions for U1 and �1 satisfy the system ofEq. �9�, and they are the first solutions of this system.

The second solution of the system of ordinary differential equa-tions with nonconstant coefficients �Eq. �9�� must be considered as

U2�r� = �A2J1��r� + A3rJ2��r��14�

�2�r� = �B2J0��r� + B3rJ1��r��

Substituting Eq. �14� into Eq. �9� yields

��2 − �2d2�A3 + d1�B3�rJ0��r� + ��− �2 + �2d2�A2 + �2d22

�A3

− d1�B2�J1��r� = 0

�15��d4�A2 + �− �2 + �d3�B2 + 2�B3�J0��r� + ��d4�A3 + �− �2

+ �d3�B3�rJ1��r� = 0

The expressions for U2 and �2 can be the solutions of Eq. �9� ifand only if

�− �2 + �2d2� − d1�

�d4� �− �2 + �d3� �A3

B3� = �0

0� �16�

�− �2 + �2d2�A2 + �2d22

�A3 − d1�B2 = 0 �17�

�d4�A2 + �− �2 + �d3�B2 + 2�B3 = 0 �18�

The nontrivial solution of Eq. �16� is obtained by setting the de-terminant equal to zero as

�− �2 + �2d2��− �2 + �d3� + d1d4�2� = 0 �19�

The characteristic Eq. �19� is the same as the characteristic Eq.�13�. This equality is interesting as it prevents mathematical di-lemma and complexity, and a single value for the eigenvalue �simultaneously satisfies both characteristic Eqs. �13� and �19�.Equations �17� and �18� give the relation between A2, A3, B2, andB3, and they play as the balancing ratios that help Eq. �14� to bethe second solution of the system of Eq. �9�. Equations �17� and�18� are two algebraic equations with six unknown A2, A3, B2, B3,�, and �. Since the number of unknowns are more than the equa-tions, there is no restriction to bring noncompatibility betweenEqs. �16�–�18�. The complete general solutions for the solid cyl-inder are

Ug�r� = A1J1��r� + A2�J1��r� + 1rJ2��r��20�

�g�r� = A12J0��r� + A2�3J0��r� + 4rJ1��r��

Those for the hollow cylinder are

Ug�r� = A1J1��r� + A2�J1��r� + 1rJ2��r�� + A3Y1��r�

+ A4�Y1��r� + 1rY2��r��21�

�g�r� = A12J0��r� + A2�3J0��r� + 4rJ1��r�� + A32Y0��r�

+ A4�3Y0��r� + 4rY1��r��

where 1–4 are ratios obtained from Eqs. �12�, �17�, and �18� andare given in Appendix A. Substituting Ug and �g in the homoge-neous form of the boundary conditions �Eq. �6��, four linear alge-
braic equations are obtained as
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wi�Ai�

wf�

Ao

w

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11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

��A1

A2

A2

A3

� = �0

0

0

0� �22�

here ij are the coefficients depending on � and � and are givenn Appendix A. Setting the determinant of the coefficients of Eq.22� equal to zero, the second characteristic equation is obtained.

simultaneous solution of this equation and Eq. �13� results in annfinite number of two eigenvalues, �n and �n. Therefore, Ug andg are rewritten as

Ug�r� = A1�J1��nr� + �5J1��nr� + 6rJ2��nr�� + 7Y1��nr�

+ �8Y1��nr� + 9rY2��nr��23�

�g�r� = A1�10J0��nr� + �11J0��nr� + 12rJ1��nr�� + 13Y0��nr�

+ �14Y0��nr� + 15rY1��nr��

here 5–15 are ratios presented in Appendix A and are obtainedrom Eq. �22�. Let us show the functions in the brackets of Eq.23� by functions H1 and H0 as

H1��nr� = J1��nr� + �5J1��nr� + 6rJ2��nr�� + 7Y1��nr�

+ �8Y1��nr� + 9rY2��nr��24�

H0��nr� = 10J0��nr� + �11J0��nr� + 12rJ1��nr�� + 13Y0��nr�

+ �14Y0��nr� + 15rY1��nr��

ccording to the Sturm–Liouville theories, these functions arerthogonal with respect to the weight function p�r�=r as

�ri

ro

H��nr�H��mr�rdr = � 0 n � m

�H��nr��2 n = m� �25�

here �H��nr�� is the norm of the H function and equals

�H��nr�� = �ri

ro

rH2��nr�dr1/2

�26�

ue to the orthogonality of function H, every piecewise continu-us function, such as f�r�, can be expanded in terms of the func-ion H �either for H0 or H1� and is called the H-Fourier series as

f�r� = �n=1

�

enH��nr� �27�

here en equals

en =1

�H��nr��2�ri

ro

f�r�H�r�rdr �28�

ccording to the numerical results, there are three groups forigenvalues �n, where the first �n1 is real and negative, and theecond and third ones, �n2 and �n3, are conjugate complex with aegative real part, −�n n, and an imaginary part, � dn. Terms dnnd n are the damped and nondamped thermal-mechanical natu-al frequencies, and �n is the damping ratio for the nth naturalode. Equation �13� is an algebraic equation in polynomial form,

nd the determinant of Eq. �22� is an algebraic equation in theessel function form. The exact analytical solution for this systemf nonlinear algebraic equations is complicated, and the numericalethod of solution is employed in this paper. Since the Bessel

unctions are periodic, the system has an infinite number of roots.he numerical results of �n and �n for 50 roots are presented inec. 4. Using Eqs. �8�, �23�, and �24�, the displacement and tem-
erature distributions due to the general solution become
51204-4 / Vol. 133, OCTOBER 2011


ug�r,t� = �n=1

�

�ane�n1t + e−� nt�bn cos dnt + cn sin dnt��H1��nr�

�29�

Tg�r,t� = �n=1

�

�ane�n1t + e−� nt�bn cos dnt + cn sin dnt��H0��nr�

Using the initial conditions �Eq. �7�� and with the help of Eqs.�27�–�29�, four unknown constants, an, bn, cn, and dn, are ob-tained.

3.2 Particular Solution With Nonhomogeneous BoundaryConditions. The general solutions may be used as proper func-tions to guess the particular solution adopted to the nonhomoge-neous parts of Eqs. �4� and �5� and the nonhomogeneous boundaryconditions �Eq. �6�� as

up�r,t� = �n=1

�

��G1n�t�J1��nr� + G2n�t�rJ2��nr�� + r2G5n�t��

�30�

Tp�r,t� = �n=1

�

��G3n�t�J0��nr� + G4n�t�rJ1��nr�� + r2G6n�t��

For solid cylinders, the second type of Bessel function Y is ex-cluded. It is necessary and suitable to expand the body forceF�r , t� and heat source Q�r , t� in the H-Fourier expansion form as

F�r,t� = �n=1

�

Fn�t�H1��nr�

�31�

Q�r,t� = �n=1

�

Qn�t�H0��nr�

where Fn�t� and Qn�t� are

Fn�t� =1

�H1��nr��2�ri

ro

F�r,t�H1��nr�rdr

�32�

Qn�t� =1

�H0��nr��2�ri

ro

Q�r,t�H0��nr�rdr

Substituting Eqs. �30� and �31� into the nonhomogeneous form ofEqs. �4� and �5� yields

�− �n2G1n�t� + d2G1n�t� +

2

�nd2G2n�t� − d1�nG3n�t� + d8d15G5n�t�

+ d8d162

�nG5n�t� + d9d15G6n�t� + d9d16

2

�nG6n�t�

+ d10d15G5n�t� + d10d162

�nG5n�t� + d5d7d15

+2

�nd5d7d16�rJ0��nr� + ��n

2G2n�t� − d2G2n�t� + d1�nG4n�t�

− d8d16G5n�t� − d10d16G5n − d4d16G6n�t� − d5d7d16�J1��nr� = 0�33�

�d4�nG1n�t� − �n2G3n�t� + d3G3n�t� + 2�nG4n�t� + d9d17G5n�t�

+ d12d17G6n�t� + d14d17G6n�t� + d6d11d17�J0��nr�

+ �d4�nG2n�t� − �n2G4n + d3G4n�t� + d13d18G5n�t�

+ d12d18G6n�t� + d13d18G6n�t� + d6d11d18�rJ1��nr� = 0

where d7–d18 and d37 and d38 are the coefficients of H-expansion
and are given in Appendix A. The guessed functions �Eq. �30��


co

EGwfn

J

Dow

an satisfy the nonhomogeneous part of Eqs. �4� and �5� if andnly if

− �n2G1n�t� + d2G1n�t� +

2

�nd2G2n�t� − d1�nG3n�t� + d8d15G5n�t�

+ d8d162

�nG5n�t� + d9d15G6n�t� + d9d16

2

�nG6n�t�

+ d10d15G5n�t� + d10d162

�nG5n�t� + d5d7d15 +

2

�nd5d7d16 = 0

�n2G2n�t� − d2G2n�t� + d1�nG4n�t� − d8d16G5n�t� − d10d16G

− d4d16G6n�t� − d5d7d16 = 0�34�

cylinder.



d4�nG1n�t� − �n2G3n�t� + d3G3n�t� + 2�nG4n�t� + d9d17G5n�t�

+ d12d17G6n�t� + d14d17G6n�t� + d6d11d17 = 0

d4�nG2n�t� − �n2G4n + d3G4n�t� + d13d18G5n�t� + d12d18G6n�t�

+ d13d18G6n�t� + d6d11d18 = 0

Taking the Laplace transform of Eq. �34� and using two boundaryconditions of Eq. �6� �for solid cylinders, only the second andfourth boundary conditions are applicable� give

0 �n

2 − d2s2 0 d1�n − d8d16 − d10d16s2 − d9d16

− �n2 + d2s2 2

�nd2S2 − d1�n 0 d19 + d20s

2 d9d15 + d9d162

�n

d4�ns 0 − �n2 + d3s 2d13d17�n d13d17s d12d17 + d14d18s

0 d4�ns 0 − �n2 + d3s d13d18s d12d18 + d14d18s

d21 d22 d23 d24 d25 d26

0 0 d27 d28 0 d29

� �35�

quation �35� is a system of algebraic equations and is solved for1n�s�–G6n�s� by the Cramer methods in the Laplace domain,here by the inverse Laplace transform the functions are trans-

ormed into the real time domain. In the process of solution, it isecessary to consider the following points.

1. The initial conditions �Eq. �7�� are considered only for thegeneral solutions �Eq. �29��. The initial conditions ofG1n�t�–G6n�t� for the particular solutions are consideredequal to zero.

2. Equation �35� is in polynomial form function of the Laplaceparameter s �not the Bessel function form of s�. Therefore,the exact inverse Laplace transform is possible and some-how simple.

3. For the hollow cylinder, it is enough to include the secondtype of the Bessel function Y�r� in the sequence of particularsolution as

up�r,t� = �n=1

�

��G1n�t�J1��nr� + G2n�t�rJ2��nr��

+ �G3n�t�Y1��nr� + G4n�t�rY2��nr�� + rG5n�t�

+ r2G6n�t��36�

Tp�r,t� = �n=1

�

��G7n�t�J0��nr� + G8n�t�rJ1��nr��

+ �G9n�t�Y0��nr� + G10n�t�rY1�t��nr�� + rG11n�t�

+ r2G12n�t��

Substituting Eq. �36� in Eqs. �4� and �5�, eight equations areobtained, where using the four boundary conditions �Eq.�6��, 12 functions G1n�t�–G12n�t� are obtained for the hollow

4 Results and DiscussionAs an example, a solid cylinder with ro=1m made of alu-

minium is considered. The material properties are shown in Table1.

Using the values of material properties given in Table 1, thefirst 50 roots of �n and �n are calculated, as shown in Table 2.

The initial temperature T� is considered to be 293 K. Now, aninstantaneous hot outside surface temperature T�1, t�=10−3T��t�,where ��t� is the unit Dirac function, is considered, and the out-side radius of the cylinder is assumed to be fixed �u�1, t�=0�. Todraw the graphs, a nondimensional time t=vt /r� is considered,where v=�E�1−�� /��1+��1−2�� is the dilatational-wave veloc-ity. Figures 1 and 2 show the wave fronts of displacement andtemperature.

As a second example, a mechanical shock wave of the formu�1, t�=10−12u��t� is applied to the outside surface of the cylin-der, where the surface is assumed to be at zero temperature�T�1, t�=0� and u�=ro�T� is an assumed initial displacement. Fig-ures 3 and 4 show the wave fronts for the displacement and tem-perature distributions along the radial direction. The convergenceof the solutions for these examples is achieved by consideration of1200 eigenvalues used for the H-Fourier expansion. More thanthis number of eigenvalues result in the increased round-off and

Table 1 Material properties of aluminum

Material properties of aluminium

E=70 GPa�=0.3�=23�10−6 1 /K�=2707 kg /m3

K=204 W /m Kc=903 J /kg K

OCTOBER 2011, Vol. 133 / 051204-5


tvtt

fscoe

5

ts

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runcation errors, which affect the quality of the graphs. The con-ergence of solution is faster for displacement in comparison withhe temperature. The small oscillations in Figs. 2 and 4 are due tohe convergence of solutions.

Now, consider a periodic temperature shock at the outside sur-ace in the form T�1, t�=T� sin t, with u�1, t�=0. Figures 5–8how the penetration waves along the radius of the cylinder. Theonvergence of solution for this example is faster than the previ-us examples, where 500 eigenvalues are used for the H-Fourierxpansion.

ConclusionIn this paper, an analytical solution for the coupled thermoelas-

icity of thick cylinders under radial temperature or mechanical

Table 2 Eigenv

oot number �n �n1

1 9.88�10−01 −1.02�10−05 −49.2882 2.10�10+00 −4.60�10−05 −223.0713 3.20�10+00 −1.07�10−04 −517.7544 4.30�10+00 −1.93�10−04 −934.9045 5.40�10+00 −3.04�10−04 −1.47446 6.50�10+00 −4.40�10−04 −2.13637 7.60�10+00 −6.02�10−04 −2.92068 8.70�10+00 −7.89�10−04 −3.82739 9.80�10+00 −1.00�10−03 −4.856310 1.09�10+01 −1.24�10−03 −6.007811 1.20�10+01 −1.50�10−03 −7.281512 1.31�10+01 −1.79�10−03 −8.677713 1.42�10+01 −2.10�10−03 −10.196314 1.53�10+01 −2.44�10−03 −11.837215 1.64�10+01 −2.80�10−03 −13.600516 1.75�10+01 −3.19�10−03 −15.486117 1.86�10+01 −3.60�10−03 −17.494218 1.97�10+01 −4.04�10−03 −19.624619 2.08�10+01 −4.51�10−03 −21.877420 2.19�10+01 −5.00�10−03 −24.252621 2.30�10+01 −5.51�10−03 −26.750122 2.41�10+01 −6.05�10−03 −29.370123 2.52�10+01 −6.62�10−03 −32.112424 2.63�10+01 −7.21�10−03 −34.977025 2.74�10+01 −7.82�10−03 −37.964126 2.85�10+01 −8.46�10−03 −41.073527 2.96�10+01 −9.13�10−03 −44.305328 3.07�10+01 −9.82�10−03 −47.659529 3.18�10+01 −1.05�10−02 −51.136030 3.29�10+01 −1.13�10−02 −54.735031 3.40�10+01 −1.20�10−02 −58.456332 3.51�10+01 −1.28�10−02 −62.300033 3.62�10+01 −1.37�10−02 −66.266034 3.73�10+01 −1.45�10−02 −70.354535 3.84�10+01 −1.54�10−02 −74.565336 3.95�10+01 −1.63�10−02 −78.898537 4.06�10+01 −1.72�10−02 −83.354038 4.17�10+01 −1.81�10−02 −87.932039 4.28�10+01 −1.91�10−02 −92.632340 4.39�10+01 −2.01�10−02 −97.455041 4.50�10+01 −2.11�10−02 −102.400042 4.61�10+01 −2.21�10−02 −107.467543 4.72�10+01 −2.32�10−02 −112.657344 4.83�10+01 −2.43�10−02 −117.969545 4.94�10+01 −2.54�10−02 −123.404046 5.05�10+01 −2.66�10−02 −128.961047 5.16�10+01 −2.77�10−02 −134.640348 5.27�10+01 −2.89�10−02 −140.442049 5.38�10+01 −3.02�10−02 −146.366150 5.49�10+01 −3.14�10−02 −152.4125

hock load is presented. The method is based on the eigenfunction

51204-6 / Vol. 133, OCTOBER 2011


Fourier expansion, which is a classical and traditional method ofsolution for the typical initial and boundary value problems. Thestrength of this method is its ability to reveal the fundamentalmathematical and physical properties and the interpretations of theproblem under study.

In the coupled thermoelastic problem of the radial-symmetriccylinder, the governing equations are a system of partial differen-tial equations with two independent variables, the radius �r� andthe time �t�. The traditional procedure to solve this class of prob-lems is to eliminate the time variable by using the Laplace trans-form. The resulting system is a set of ordinary differential equa-tions in terms of the radius variable, which falls into the Besselfunction family. This method of analysis brings the Laplace pa-rameter �s� in the argument of the Bessel functions, causing hard-

s for fifty roots

�n2 �n3

10−009+5.7338�10+003i −49.2889�10−009−5.7338�10+003i

10−009+12.1963�10+003i −223.0715�10−009−12.1963�10+003i

10−009+18.5802�10+003i −517.7543�10−009−18.5802�10+003i

10−009+24.9668�10+003i −934.9049�10−009−24.9668�10+003i

0−006+31.3536�10+003i −1.4744�10−006−31.3536�10+003i

0−006+37.7403�10+003i −2.1363�10−006−37.7403�10+003i

0−006+44.1272�10+003i −2.9206�10−006−44.1272�10+003i

0−006+50.5140�10+003i −3.8273�10−006−50.5140�10+003i

0−006+56.9008�10+003i −4.8563�10−006−56.9008�10+003i

0−006+63.2876�10+003i −6.0078�10−006−63.2876�10+003i

0−006+69.6744�10+003i −7.2815�10−006−69.6744�10+003i

0−006+76.0613�10+003i −8.6777�10−006−76.0613�10+003i

0−006+82.4481�10+003i −10.1963�10−006−82.4481�10+003i

0−006+88.8349�10+003i −11.8372�10−006−88.8349�10+003i

0−006+95.2217�10+003i −13.6005�10−006−95.2217�10+003i

0−006+101.6086�10+003i −15.4861�10−006−101.6086�10+003i

0−006+107.9954�10+003i −17.4942�10−006−107.9954�10+003i

0−006+114.3822�10+003i −19.6246�10−006−114.3822�10+003i

0−006+120.7690�10+003i −21.8774�10−006−120.7690�10+003i

0−006+127.1559�10+003i −24.2526�10−006−127.1559�10+003i

0−006+133.5427�10+003i −26.7501�10−006−133.5427�10+003i

0−006+139.9295�10+003i −29.3701�10−006−139.9295�10+003i

0−006+146.3163�10+003i −32.1124�10−006−146.3163�10+003i

0−006+152.7032�10+003i −34.9770�10−006−152.7032�10+003i

0−006+159.0900�10+003i −37.9641�10−006−159.0900�10+003i

0−006+165.4768�10+003i −41.0735�10−006−165.4768�10+003i

0−006+171.8636�10+003i −44.3053�10−006−171.8636�10+003i

0−006+178.2505�10+003i −47.6595�10−006−178.2505�10+003i

0−006+184.6373�10+003i −51.1360�10−006−184.6373�10+003i

0−006+191.0241�10+003i −54.7350�10−006−191.0241�10+003i

0−006+197.4109�10+003i −58.4563�10−006−197.4109�10+003i

0−006+203.7978�10+003i −62.3000�10−006−203.7978�10+003i

0−006+210.1846�10+003i −66.2660�10−006−210.1846�10+003i

0−006+216.5714�10+003i −70.3545�10−006−216.5714�10+003i

0−006+222.9582�10+003i −74.5653�10−006−222.9582�10+003i

0−006+229.3451�10+003i −78.8985�10−006−229.3451�10+003i

0−006+235.7319�10+003i −83.3540�10−006−235.7319�10+003i

0−006+242.1187�10+003i −87.9320�10−006−242.1187�10+003i

0−006+248.5055�10+003i −92.6323�10−006−248.5055�10+003i

0−006+254.8924�10+003i −97.4550�10−006−254.8924�10+003i

0−006+261.2792�10+003i −102.4000�10−006−261.2792�10+003i

0−006+267.6660�10+003i −107.4675�10−006−267.6660�10+003i

0−006+274.0528�10+003i −112.6573�10−006−274.0528�10+003i

0−006+280.4397�10+003i −117.9695�10−006−280.4397�10+003i

0−006+286.8265�10+003i −123.4040�10−006−286.8265�10+003i

0−006+293.2133�10+003i −128.9610�10−006−293.2133�10+003i

0−006+299.6001�10+003i −134.6403�10−006−299.6001�10+003i

0−006+305.9869�10+003i −140.4420�10−006−305.9869�10+003i

0−006+312.3738�10+003i −146.3661�10−006−312.3738�10+003i

0−006+318.7606�10+003i −152.4125�10−006−318.7606�10+003i

alue

9�5�3�9��1�1�1�1�1�1�1�1�1�1�1

�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1�1

ship or complications in carrying out the exact inverse of the



LLopstis

FT

FT

Fu

J

Dow

aplace transformation. As a result, the numerical inverse of theaplase transformation is used in the papers dealing with this typef problems in literature. In the present paper, to prevent thisroblem, when the Laplace transform is applied to the particularolutions, it is postponed after eliminating the radius variable r byhe H-Fourier expansion. Thus, the Laplace parameter �s� appearsn polynomial function forms, and hence the exact Laplace inver-ion transformations are possible.

ig. 1 Nondimensional displacement distribution due to input„1, t…=10−3TŒ�„t… at nondimensional time t=0.4

ig. 2 Nondimensional temperature distribution due to input„1, t…=10−3TŒ�„t… at nondimensional time t=0.4

ig. 3 Nondimensional displacement distribution due to input−12 ˆ

„1, t…=10 uŒ�„t… at nondimensional time t=0.4



The method described in this paper is an exact solution of thecoupled thermoelasticity of thick cylinders with two given bound-ary conditions. The exact solutions found in literature for thecoupled thermoelasticity problems are limited to the infinitespaces and half-spaces. The analytical method of solution pre-sented in this paper is for a finite domain with specified and givenboundary conditions. This is the novelty of the paper, where thesolution of a popular structural component �a thick cylinder� with

Fig. 4 Nondimensional temperature distribution due to inputu„1, t…=10−12uŒ�„t… at nondimensional time t=0.4

Fig. 5 Nondimensional temperature distribution due to inputT„1, t…=TŒ sin t at nondimensional time t=0.3

Fig. 6 Nondimensional temperature distribution due to inputˆ ˆ
T„1, t…=TŒ sin t at nondimensional time t=0.6
OCTOBER 2011, Vol. 133 / 051204-7


tts

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wo specified boundary conditions under the coupled thermoelas-ic assumption is given analytically and in terms of the seriesolution.

ppendix ASee the following parameters:

d1 = − ��1 + ��1 − ��

d2 = − ��1 + ��1 − 2��

E�1 − ��

d3 =�c

k

d4 = −E�

k�1 − 2��

d5 =�1 + ��1 − 2��

E�1 − ��

d6 =1

k

d7 =�0

1

F�r�d34rdr

d8 =�0

1

3d34rdr

d9 =�0

1

2d1d34r2dr

d10 =�0

1

d2d34r3dr

d11 =�0

1

Q�r�d35rdr

d12 =�1

4d35rdr

ig. 7 Nondimensional displacement distribution due to input„1, t…=TŒ sin t at nondimensional time t=0.3

0

51204-8 / Vol. 133, OCTOBER 2011


d13 =�0

1

3d4d35r2dr

d14 =�0

1

d3d35r3dr

d15 = ��− J1��n�d2d3�n3 + J1��n��4 + 1

2�n5J1��n� − 1

2�n3J2��n�d3�n

− 12�n

3J2��n�d2�n2 + 1

2�nJ2��n�d2�n3d3

+ 12�n

3J2��n��nd4d1��/�J1��n��d2�n3d3 − �n

4� + 1�

d16 = − 12�n��n

4 − �n2d3� − d2�n

2�n2 + d2�n

3d3 + �nd4�n2d1�/�d2�n

3d3

− �n4�

d17 = J1��n��4 + 12�n

5J1��n�

d18 = �nd4�n2d1

d19 = d8�d15 + d162

�n�

d20 = d10�d15 + d162

�n�

d21 = C21J1��nr�� + C22�J0��nr�� −1

r�

C22J1��nr��

d22 = C21r�J2��nr�� − C22J2��nr�� + C22�r�J1��nr��

d23 = C23J0��nr��

d24 = C23r�J1��nr��

d25 = C21r�2 + 2C22r�

d26 = C23r�2

d27 = C41J0��nr�� − C42�J1��nr��

d28 = C41r�J1��nr�� + C42�r�J0��nr��

d29 = C41r�2 + 2C42r

Fig. 8 Nondimensional displacement distribution due to inputT„1, t…=TŒ sin t at nondimensional time t=0.6

d30 = d5d7d16



J

Dow

d31 = − d5d7d15 −1

�nd5d7d16

d32 = − d6d11d17

d33 = − d6d11d18

d34 = d15J1��nr� + d16rJ2��nr�

d35 = d17J0��nr� + d18rJ1��nr�

2 =− �n

2 + �2d2

d1�n

3 = 2 +2�2d2

�n2d1

1

4 =1

2�n��d4�n + �− �n

2 + �d3�3�

5 = 18

6 = 118

7 = 17

8 = 16

9 = 116

10 = 2

11 = 318

12 = 418

13 = 217

14 = 316

15 = 416

16 =

�−31

32+

41

42� −

�−33

32+

43

42��21 −

2211

12�

�−2213

12+ 23�

�−34

32+

44

42� −

�−33

32+

43

42��24 −

2214

12�

�−2213

12+ 23�

17 = −�21 −

2211

12�

�23 −2213

12� −

�24 −2214

12�

�23 −2213

12� 16

18 = −11

12−

13

1217 −

14

1216

11 = C11J1��nri� + C12��nJ0��nri� −1

J1��nri�� + C132J0��nri�
ri


12 = C11�J1��nri� + 1riJ2��nri�� + C12��nJ0��nri� −1

riJ1��nri�

+ 1J2��nri� + 1ri�nJ1��nri� − 21J2��nri�� + C13�3J0��nri�

+ 4riJ1��nri��

13 = C11Y1��nri� + C12��nY0��nri� −1

riY1��nri�� + C132Y0��nri�

14 = C11�Y1��nri� + 1riY2��nri�� + C12��nY0��nri� −1

riY1��nri�

+ 1Y2��nri� + 1ri�nY1��nri� − 21Y2��nri��+ C13�3Y0��nri� + 4riY1��nri��

21 = C21J1��nr�� + C22��nJ0��nr�� −1

r�

J1��nr�� + C232J0��nr��

22 = C21�J1��nr�� + 1r�J2��nr�� + C22��nJ0��nr�� −1

r�

J1��nr��

+ 1J2��nr�� + 1r��nJ1��nr�� − 21J2��nr��+ C13�3J0��nr�� + 4r�J1��nr��

23 = C21Y1��nr�� + C22��nY0��nr�� −1

r�

Y1��nr�� + C232Y0��nr��

24 = C21�Y1��nr�� + 1r�Y2��nr�� + C22��nY0��nr�� −1

r�

Y1��nr��

+ 1Y2��nr�� + 1r��nY1��nr�� − 21Y2��nr��+ C13�3Y0��nr�� + 4r�Y1��nr��

31 = C312J0��nri� − C322�nJ1��nri�

32 = C31�3J0��nri� + 4riJ1��nri�� + C32�− 3�nJ1��nri�

+ 4J1��nri� + 4ri�nJ0��nri� − 4J1��nri��

31 = C312Y0��nri� − C322�nY1��nri�

32 = C31�3Y0��nri� + 4riY1��nri�� + C32�− 3�nY1��nri�

+ 4Y1��nri� + 4ri�nY0��nri� − 4Y1��nri��

31 = C312J0��nr�� − C322�nJ1��nr��

32 = C31�3J0��nr�� + 4r�J1��nr�� + C32�− 3�nJ1��nr��

+ 4J1��nr�� + 4r��nJ0��nr�� − 4J1��nr��

31 = C312Y0��nr�� − C322�nY1��nr��

32 = C31�3Y0��nr�� + 4r�Y1��nr�� + C32�− 3�nY1��nr��

+ 4Y1��nr�� + 4r��nY0��nr�� − 4Y1��nr��

Appendix B: Mechanical Boundary ConditionsFor mechanical boundary conditions, four options are available:

known radial displacement, known radial stress, and a combina-tion of them.

�1� Radial displacements at inner and outer surfaces are known
as
OCTOBER 2011, Vol. 133 / 051204-9


z

cd

A

s

e

cd

R

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u�ri,t� = f1�t�

u�ro,t� = f2�t�

In this case, we have C11=1, C12=0, C21=1, and C22=0.For fixed surfaces, it is enough to consider f1�t� and f2�t�equal to zero.

�2� When radial stress at inner and outer surfaces are known,by the help of Eqs. �1� and �2�, we can write

�rr�ri,t� =E

�1 + ��1 − 2��1 − ��u,r + �1

riu

−E�

�1 − 2��T�ri,t� = f1�t�

�rr�ro,t� =E

�1 + ��1 − 2��1 − ��u,r + �1

rou

−E�

�1 − 2��T�ro,t� = f2�t�

In this case, we have C11= E��1+��1−2�� , C12=

E�1−��

�1+��1−2�� , C21

= E��1+��1−2�� , and C22=

E�1−��

�1+��1−2�� .

For traction free, it is enough to consider f1�t� and f2�t� equal toero.

The third and fourth mechanical boundary conditions are theombination of above mentioned first and second boundary con-itions.

ppendix C: Thermal Boundary ConditionsFor thermal boundary conditions, six options are available:

pecified temperature, heat flux, and convection.

�1� Temperatures at inner and outer surfaces are known as

T�ri,t� = f1�t�

T�ro,t� = f2�t�

In this case, we have C11=1, C12=0, C21=1, and C22=0.�2� Heat fluxes at inner and outer surfaces are known as

T,r�ri,t� = f1�t�

T,r�ro,t� = f2�t�

In this case, we have C31=k, C32=0, C41=k, and C42=0,where k is the thermal conduction coefficient.

For an insulated surface, it is enough to consider f1�t� and f2�t�qual to zero.

�3� Convections at inner and outer surfaces are known as

hiT�ri,t� + kT,r�ri,t� = f3�t�

hiT�ro,t� + kT,r�ro,t� = f4�t�

In this case, we have C31=hi, C32=k, C41=ho, and C42=k,where hi and ho are the thermal convection coefficients atinner and outer surfaces of the cylinder, respectively.

The fourth to sixth cases for thermal boundary conditions areombinations of the above-mentioned first to third boundary con-itions.

eferences�1� 2007, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1,

ASME, New York.
�2� Hetnarski, R. B., 1964, “Solution of the Coupled Problem of Thermoelasticity
51204-10 / Vol. 133, OCTOBER 2011


in the Form of Series of Functions,” Arch. Mech. Stosow., 16, pp. 919–941.�3� Hetnarski, R. B., and Ignaczak, J., 1993, “Generalized Thermoelasticity:

Closed-Form Solutions,” J. Therm. Stresses, 16, pp. 473–498.�4� Hetnarski, R. B., and Ignaczak, J., 1994, “Generalized Thermoelasticity: Re-

sponse of Semi-Space to a Short Laser Pulse,” J. Therm. Stresses, 17, pp.377–396.

�5� Georgiadis, H. G., and Lykotrafitis, G., 2005, “Rayleigh Waves Generated by aThermal Source: A Three-Dimensional Transient Thermoelasticity Solution,”ASME J. Appl. Mech., 72, pp. 129–138.

�6� Wagner, P., 1994, “Fundamental Matrix of the System of Dynamic LinearThermoelasticity,” J. Therm. Stresses, 17, pp. 549–565.

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