Thermal stresses in an infinite circular cylinder

Journal of Mechanical Science and Technology 26 (6) (2012) 1829~1839

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0424-5

Thermal stresses in an infinite circular cylinder†

A. M. Abd-Alla1, G. A.Yahya2,3 and A. M. Farhan4 1Department of Mathematics, Faculty of Science, Taif University, Saudi Arabia

2Department of Physics, Faculty of Science, Taif University, Saudi Arabia 3Department of Physics, Faculty of Science, Aswan, South Valley University, Egypt

4Department of Physics, Faculty of Science, Jazan University, Saudi Arabia

(Manuscript Received October 17, 2011; Revised February 1, 2012; Accepted February 10, 2012)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In the present paper, the problem of dynamic thermal stresses in an infinite elastic cylinder of radius a, with its axis along the z-axis,

subject to certain boundary conditions, is studied. This type of situation can arise due to melting at constant rate of an insulating material at zero temperature deposited on the positive half of the infinite cylinder. A solution and numerical results are obtained for the stress components, displacement components, and temperature. Numerical results are given and illustrated graphically for each case considered.

Keywords: Wave propagation; Thermoelastic solid; Isotropic material; Thermoelasticity, Thermal stress ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

In recent years, owing to the diversifications of engineering materials and to the operations at severe thermal environments, the isotropic thermal stress problems have become very im-portant to modern designs such as nuclear reactors. The ther-mal stress problem in transversely isotropy which is a kind of anisotropy have been studied by many researches such as Noda et al. [1], Tsai [2], and Chandrasekharaiah and Kesha-van [3]. Generalized thermoelastic infinite medium with cy-lindrical cavity subjected to moving heat source has been in-vestigated by Youssef [4]. Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section is stud-ied by Ponnusamy [5]. The rotation of non-homogeneous composite infinite cylinder were investigated by El-Naggar et al. [6], by considering an orthotropic cylinder containing (i) an isotropic core and (ii) a rigid core. Shama and Grover [7] dis-cussed body wave probagation in rotating thermoelastic media. Wave probagation in a generalizad thermoelastic solid cylin-der of arbitrary cross-section immersed in a fluid have em-ployed by Venkatesan and Ponnusamy [8]. Kardomateas [9] investigated transient thermal stresses in cylindrically orthotropic composite tubes. Green and Lindsay [10] dis-cussed the thermoelasticity. Abd-Alla and Abo-Dahab [11] investigated time-harmonic in a generalized magneto-thermo-viscoelastic continum with and without energy dissipation. Body wave propagation in rotating elastic media has been

investigated by Auriault [12]. Methods of theoretical physics have been employed by Morse and Feshbach [13]. Elastic constants of anisotropic material has been introduced by Hearmon [14]. Dhaliwal and Choudhary [15] have employed methods of integral transforms and variation of parameters to solve dynamic thermoelastic problem for cylindrical regions. Abd-Alla et al. [16] have solved the Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic mate-rial under initial stress and gravity field. A.M. Abd-Alla, et al. [17] investigated magneto-thermo-viscoelastic interactions in an unbounded body with a spherical cavity subjected to a pe-riodic loading,. Kumar and Mukhopadhyay [18] studied of the effects of thermal relaxation time on plane wave propagation under two-temperature thermoelasticity. Abd-Alla and Mah-moud [19] investigated magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model.

In this paper a general analytical solution using the tech-nique of variables separation on the basis of linear theory of elasticity is developed with boundary conditions. The thermal stresses in an infinite cylinder of radius a due to a constant temperature applied to a variable portion of the curved surface while the rest of surface is maintained at zero temperature is studied. Such situation can arise due to the melting of an insu-lating material deposited on the surface of the cylinder. The heat conduction equation has been solved by applying the Fourier transform and the theory of complex variables. The thermoelastic equation of motion has been split into two wave equations representing rotational and irrotational displacement. The particular integral of the longitudinal wave equation has

*Corresponding author. Tel.: +966545087011, Fax.: +96627330615 E-mail address: [email protected]

† Recommended by Associate Editor Jun-Sik Kim © KSME & Springer 2012

1830 A. M. Abd-Alla et al. / Journal of Mechanical Science and Technology 26 (6) (2012) 1829~1839

been obtained in the form of series involving Bessel functions of the first kind and of order zero. Numerical results are pre-sented for the variation of the displacement and stresses with time and through the radial.

2. Formulation of the problem

Let us consider a homogeneous and isotropic elastic solid (infinite circular cylinder of radius a, whose axis is along the z-axis, subject to the certain boundary conditions. The corre-sponding components of the displacement vector U at a point are rU and zU in the r and z directions, respectively.

The boundary condition for the temperature is

(1)

where t is the time and ν is a positive constant. The condition of form (1) can arise due to melting at a con-

stant rate of an insulating material at zero temperature depos-ited on the positive half-space of the infinite cylinder.

The curved surface of the cylinder is assumed to be fixed (i.e.):

(2)

where Ur and Uz is the displacement components. With assumption that:

. (3)

Due to the rotational symmetry of the cylinder which has

been considered in the present study, all field quantities are independent of the polar angle θ .

Introducing the non-dimensional quantities:

. The classical equations of dynamical thermoelasticity of the

a homogeneous isotropic elastic body in the absence of body forces, in terms of displacement vector and the temperature in the non-dimensional form are given by [15]:

(4)

. (5) The above equations represent thermoelastic equation and

heat conduction equation respectively, where U is the dis-placement vector, T is the temperature change from the equi-librium temperature T0, k is the coefficient of thermal conduc-tivity, ρ is the density, Ce is the specific heat, α is the coeffi-cient of thermal expansion and λ, µ are Lamr's constants.

The boundary condition (1) in the non-dimensional form becomes

.

(6)

In the following discussion the primes have been neglected.

3. Solution of the heat conduction equation

Applying the Fourier transform:

to Eqs. (5) and (6) and using Eq. (3) we get:

(7)

and for r =1; t > 0

. (8)

Eq. (7) under the boundary condition (8) has a solution in

the form:

(9)

where 1( )f r and 2( )f r satisfy the following differential equations:

(10)

(11)

subject to the boundary conditions:

. (12)

Eqs. (10) and (11) are satisfied by the solutions:

(13)

A. M. Abd-Alla et al. / Journal of Mechanical Science and Technology 26 (6) (2012) 1829~1839 1831

where 0I is the modified Bessel function of first kind and order zero.

Putting r =1 in Eq. (13) and using (12) we get

.

Substituting for 1 2( ), ( )f r f r in Eq. (9) the complete solu-

tion for Eq. (7) is given by

. (14)

Applying the inversion formula for Fourier’s transform

which is given by

to Eq. (14) one can get:

. (15)

The two integrals involved in Eq. (15) will be evaluated sepa-rately. By applying Jordon's lemma and Residue theorem [13] in Eq. (15) we have: 3.1 Firstly for 0z

. (16) In evaluating the first integral in Eq. (15) two cases arise

according to (t-z) is negative or positive. When (t-z) is nega-tive the integral becomes:

(17)

where , 1.n ≥ When (t-z) is positive we have,

(18)

where , 1,n ≥ nα are the positive ordered roots of the equation 0( ) 0,nJ α = 1( ),nJ α

1( )nJ α are Bessel functions of the first kind, of order zero and first order. 3.2 Secondary for 0z≺

Proceeding as before one can get from Eq. (15):

. (19) When (t-z) > 0, the result (18) holds for the case when z < 0. Using Eqs. (16)-(19) in (15) one can get:

(20)

for z > t

(21)

for 0 ≤ z ≤ t

(22)

for z < 0

4. Solution of the thermoelastic equation

By Helmholtz's theorem [13], the displacement vector U can be written in the form :

(23)

where the two functions ,φ ψ are known in the theory of elasticity, by Lamé potentials representing irrotational and rotational parts of the displacement vector U respectively. Substituting Eq. (23) into Eq. (4) we obtain:

(24)

(25)

where .

It is possible to take only one component of the vector ψ to be non-zero i.e. ψ can be written as

where r rψψ ∂

=∂

and ψ satisfies the wave Eq. (25) and

therefore has the general form:


(26)

where , ,n n nD Fγ are constants to be determined from the boundary conditions.

To obtain the particular integral of Eq. (24), T is eliminated from Eqs. (5) and (24) and we get:

. (27)

Now one can take 1 2 ,φ φ φ= + where 1φ and 2φ satisfy

the equations:

(28)

.

(29)

Also φ is the solution of Eq. (24) therefore :

.

(30)

2φ can be determined by subtracting Eq. (30) from Eq.

(29) as follows:

(31)

where 21

2cbk

= .

This differential Eq. (31) has a solution in the form:

(32) To satisfy the physical conditions in the above equation

where b2 > 0 must satisfy the following condition

. Now 2φ being a solution of Eq. (29), substituting from Eq.

(32) into Eq. (29) one can get:

which has a solution in the form:

(33)

where nA and nβ are arbitrary constants. For 1,φ the solution of Eq. (28) can be written in the form:

(34)

where , , ,n n n nB e d b and nδ are arbitrary constants. Combining Eqs. (32), (33) and (34) one can get the com-

plete solution of Eq. (24) to be:

(35) The displacement components in terms of φ and ψ are:

(36)

By using the stress-strain relations

where from which one can obtain:

(37) In order to determine the arbitrary constants in Eqs. (26)

and (35), the boundary conditions must be studied in some detail.

5. Boundary conditions

Case A: when z t

In this case it can be obtained from Eqs. (20) and (35):

.

.

.


(38) By substituting for φ from Eq. (38) and for ψ from Eq.

(26) and applying the boundary conditions (2), we get:

(39)

By taking

and equating the coefficient of exp( ),n zα− exp( )nt zα− and exp( ( ))n t zη − to zero one gets from the above equations, the displacement and stress components are given by:

(40)

(41)

(42)

Case B: when 0 z t≤ ≤

In this case using the same technique as in case A, we have:

(43)

where 2 212 ,n nB mC B= 2 2

12 ,n nD mC D=

.


From the above equations, the displacement and stress

components are given by:

(44)

Case C: when 0z≺

In this case by using the same technique as in case A and B, one can write

(45)

where 2 312 ,n nB mC B= 2 3

12 ,n nD mC D=

So that the displacement and stress components are given by:

.

.


(46)

6. Numerical results and discussion

Calculation results for the response of the non-dimensional temperature, displacement, and stresses are carried out along the r-direction at different values of time ( t = 0.25, 0.50 and 0.75) in three cases, for z t , 0 z≤ t≤ and 0z≺ are shown in Figs. 1-21 respectively. The copper material is cho-sen for numerical evaluations. In the calculations process, the material constants necessary to be known are: λ = 7.76 x 1010 Nm-2, μ = 3.86 x 1010 Nm-2, α = 1.78 x 10-5

deg-1, Cv =383.1 Jkg-1deg-1, ρ = 8954 kg/m3. Figs. 1-3 illustrate the temperatures increases with increas-

ing t . In the case z t it decreases with increasing r. Also, in the case 0 z≤ t≤ it increases with r , while it decreases with increasing r in the case 0z≺ . It can be found that from Figs. 1-3 that the temperature has a non-zero value only in a bounded region of space at a given instant. Outside this region the value vanishes and this means that the region has not felt thermal disturbance yet. At different instants, the non-zero region moves forward correspondingly with the passage of time. This indicates that heat propagates as a wave with velocity in medium. It is completely different from the case for the classical theories of thermoelasticity where an infinite speed of propagation in inherent and hence all the considered

functions have a non-zero (although may be very small) value for any point in the medium.

Figs. 4-6 illustrated the radial components of displacement. Fig. 4 shows that in the case z t , the displacement de-creases with increasing r and then it starts to increases at the value of r = 4.5, while Figs. 5, 6 show that in the case 0 z≤ t≤ and 0z≺ the displacement increases with in-creasing t and r , respectively.

Also, Figs. 7-9 show the axial displacement, in the cases z t , 0 z≤ t≤ and 0z≺ it increase with increasing r. It can be observed that the medium along the radial r adjoining

Fig. 1. Variation of the temperature T with the radial part for .z t

Fig. 2. Variation of the temperature T with the radial part for 0 .z t≤ ≤

Fig. 3. Variation of the temperature T with the radial part for 0.z≺


the surface undergoes expansion deformation because of thermal shock while the others compressive deformation. The deformation is a dynamic process. With the passage of time, the expansion region moves inside gradually and becomes larger. Thus the radial and axial displacement become larger and larger. At a given instant, the radial and axial displace-ment are finite, which is due to the wave effect of heat. It indi-cates that heat transfers into the radial of the cylinder with velocity with time passing. The more considered instant, the more the thermal stresses and, radial and axial displacement correspondingly. Due to heat propagation with velocity, it can

be also found from Figs. 4-6 that the radial and axial dis-placement are non-zero only in a cylindrical region at a given instant, that is, the thermal stresses is finite.

Figs. 10-12 illustrate the radial stress and it is shows that the stress decreases with increasing r and increases with increas-ing t in the cases z t , 0 z≤ t≤ and 0z≺ except for negative values of it.

Figs. 13-15 illustrate the tangential stress and it is shows that the stress decreases with increasing r in cases z t , 0 z≤ t≤ and 0z≺ , while, in case z t it starts to de-creases for negative values of it.

Figs. 16-18 illustrate the axial shear stress and it is shows that the stress decreases with increasing r and t and then it starts to increase at the value of r = 4.5 in case z t , while

Fig. 4. Variation of the radial component of the displacement Ur for

.z t

Fig. 5. Variation of the radial component of the displacement Ur for 0 z≤ .t≤

Fig. 6. Variation of the radial component of the displacement Ur for

0.z≺

Fig. 7. Variation of the axial component of the displacement Uz for

.z t

Fig. 8. Variation of the axial component of the displacement Uz for 0 z≤ .t≤

Fig. 9. Variation of the axial component of the displacement Uz for

0.z≺


in the cases 0 z≤ t≤ and 0z≺ , the stresses decrease with increasing r and t and it starts to increase at the value of r = 8.5.

Figs. 19-21 illustrate the axial stress and it is shows that the stress increases with increasing r in the case z t , while in the cases 0 z≤ t≤ and 0z≺ it decreases with increasing r .

From Figs. 10-12, Figs. 13-15, Figs. 16-18, and Figs. 19-21, it is easy to find the thermal stresses heat effects. The thermal stresses is placed in an initial heat field, and it deforms be-cause of thermal shock. Thus there results thermal stresses in the medium. It also can be found the thermal stresses vary with heat wave transferring into the radial of the cylinder.

Figs. 6, 17 and 18, show variation of the radial stress in a

thermoelastic cylinder. From both Figures, the radial stress increases with increasing radial r for different values of time, and it is increases with increasing time. Figs. 19, 20 and Fig. 21, respectively, show the variation of the axial stress in ther-moelastic medium. From both Figures, the axial stress in-creases with increasing radial r and it is increases with increas-ing the time t .

The variations of the stresses σrr , σθθ , σzz and τrz are due to the effect of inertia. It can be seen that the components of dis-placement Ur , Uz and the temperature T satisfy the boundary conditions. It is evident that thermal mechanic has a signifi-cant influence on the stresses. These results are specific for the

Fig. 10. Variation of the radial component of the stress σrr for .z t

Fig. 11. Variation of the radial component of the stress σrr for 0 z≤ .t≤

Fig. 12. Variation of the radial component of the stress σrr for 0.z≺

Fig. 13. Variation of the tangential component of the stress σθθ for

.z t

Fig. 14. Variation of the tangential component of the stress σθθ for 0 .z t≤ ≤

Fig. 15. Variation of the tangential component of the stress σθθ for

0.z≺


example considered, but other examples may have different trends because of the dependences of the results on the ther-mal and thermal constants of the material.

7. Conclusion

Due to the complicated nature of the governing equations for the thermoelastic theory, the work done in this field is unfortunately limited in number. The method used in this study provides a quite successful approach in dealing with such problems. This approach gives exact solution without any assumed restriction on the actual physical quantities that

appear in the governing equations of the problem considered. Important phenomena are observed in all these computations:

(1) By comparing Figs. 4, 5, 7, 8, and 9, it was found that Ur and Uz have the same behavior in both media. But the values of Ur and Uz in thermoelastic medium are larger in compari-son with those in elastic medium. The same remark for the stresses in comparing Figs. 6, 17-20, and 21.

(2) The results presented in this paper should prove useful for researchers in material science, designers of new materials, low temperature physicists as well as for those working on the

Fig. 16. Variation of the axial shear component of the stress τrz for

.z t

Fig. 17. Variation of the axial shear component of the stress τrz for 0 z≤ .t≤ .

Fig. 18. Variation of the axial shear stress component of the displace-ment τrz for 0.z≺

Fig. 19. Variation of the axial component of the stress σzz for .z t .

Fig. 20. Variation of the axial component of the stress σzz for 0 z≤ .t≤

Fig. 21. Variation of the axial component of the stress σzz for 0.z≺


development of a theory thermoelasticity. (3) It is concluded from the above analyses and results that

the present solution is accurate and reliable, and the method is simple and effective. So it may be used as a reference to solve other transient problems of uncoupled thermoelasticity.

References

[1] N. Noda, A. Shizuoka and F. A. Ashida, Three-dimensional treatment of transient thermal stress in a transversely iso-tropic semi-infinite circular cylinder subjected to a symmet-ric temperature on the cylindrical surface, Acta Mechanica, 58 (1986) 175-191.

[2] Y. M. Tsai, Thermal stress in a transversely isotropic me-dium containing a penny-shaped crack, ASME J. Appl. Mech., 50 (1993) 24-28.

[3] D. S. Chandrasekharaiah and H. R. Keshavan, Thermoelastic plane waves in a transversely isotropic body, Acta Mehanica, 47 (1991) 11-23.

[4] H. M. Youssef, Generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source, Me-chanics Research Communication, 36 (2009) 487-496.

[5] P. Ponnusamy, Wave propagation in a generalized thermoe-lastic solid cylinder of arbitrary cross-section, Int. J. of solid and structures, 44 (2007) 5336-5348.

[6] A. M. El-Naggar, A. M. Abd-Alla and S. M. Ahmed, On the rotation of a non-homogeneous composite infinite cylinder of orthotropic material, Appl. Math. and computation, 69 (1995) 174-157.

[7] J. N. Sharma and D. Grover, Body wave propagation in rotating thermoelastic media, Mechanics Research Commu-nications, 36 (2009) 715-721.

[8] M. Venkatesan and P. Ponnusamy, Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section immersed in a fluid, Int. J. of Mechanical Science, 49 (2007) 741-751.

[9] G. A. Kardomateas, Transient thermal stress in cylindrically orthotropic compsite tubes, ASME J. Appl. Mech., 56 (1989) 411-416.

[10] A. E. Green and K. A. Lindsay, Thermoelasticty, J. of Elas-ticity, 2 (1972) 1-7.

[11] A. M. Abd-Alla and S. M. Abo-Dahah, Time-harmonic in a generalized magneto-thermo-viscoelastic continum with and without energy dissipation, Applied Mathematical Modelling, 33 (2009) 2388-2402.

[12] J. L. Auriault, Body wave propagation in rotating elastic media, Mech. Res. Comm., 31 (2004) 21-27.

[13] P. M. Morse and H. Feshbach, Methods of theoretical Physics, Part I, McGraw-Hill, New York, USA (1953).

[14] R. F. S. Hearmon, Elastic constants of anisotropic materials, Rev. Phys., 18 (1946) 325-350.

[15] R. S. Dhaliwal and K. l. Choudhary, Theormoelastic prob-lem for cylindrical regions, Arch. Mech. Stos., 20 (1968) 47-65.

[16] A. M. Abd-Alla, S. M. Abo-Dahab and H. A. H. Hammad, Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field, Applied Mathematical Modeling, 35 (2011) 2981-3000.

[17] A. M. Abd-Alla, H. A. H. Hammad and S. M. Abo-Dahab, Magneto-thermo-viscoelastic interactions in an unbounded body with a spherical cavity subjected to a periodic loading, Applied Mathematic sand computation, 155 (2004) 235-248.

[18] R. Kumar and S. Mukhopadhyay, Effects of thermal relaxa-tion time on plane wave propagation under two-temperature thermoelasticity, Int. J. of Eng. Sc., 48 (2010) 128-139.

[19] A. M. Abd-Alla and S. R. Mahmoud, Magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylinder under the hyperbolic heat con-duction model, Meccanica, 45 (2010) 451- 461.

Gamal El-Deen Abdel-Raheem Yahya is currently at the Physics Department, Faculty of Science, Taif University. He has more than forty research papers pub-lished in different journals in the same field.

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Thermal stresses in an infinite circular cylinder

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