Meccanica

, Volume 47, Issue 5, pp 1295–1306

# On problem of transient coupled thermoelasticity of an annular fin

• A. M. Abd-Alla
• S. R. Mahmoud
• S. M. Abo-Dahab
Article

## Abstract

In this paper, the radial deformation and the corresponding stresses in a homogeneous annular fin for an isotropic material has been investigated. A numerical technique is proposed to obtain the solution of the transient coupled thermoelasticity in an annular fin cylinder with it’s base suddenly subject to a heat flux of a decayed exponential function of time. The system of fundamental equations is solved by using an implicit finite-difference method. The present method is a second-order accurate in time and space and unconditionally stable. A numerical method is used to calculate the temperature, displacement and the components of stresses with time t and through the radial of the annular fin cylinder. The results indicate that the effect of coupled thermoelasticity on temperature, stresses and displacement is very pronounced. Comparison is made with the results predicted by the theory of thermoelasticity in the absence of coupled thermoelasticity.

## Keywords

Coupled problem thermoelasticity Implicit finite difference method Stresses Annular fin problem

## Nomenclature

a,b

Inner and outer radii of the fin.

c

Specific heat of material of the fin.

E

Young’s modulus of material of the fin.

G

Dimensionless parameter.

h

Heat transfer coefficient.

k

Is the coefficient of the thermal conductivity.

M

qa

Heat flux from the base of the fin.

r,R

Dimension and dimensionless radial coordinate.

τ,t

Dimension and dimensionless time.

u,U

Dimension and dimensionless radial component of displacement.

β

Linear thermal expansion coefficient of material of the fin.

θ,T,T

Dimension, dimensionless and ambient temperature.

δ

Thickness of the fin.

η

It is a parameter of the thermo mechanical coupling.

ξ

Exponent of the decayed heat flux.

ρ

Density of material of the fin.

σrr,σRR

Dimension and dimensionless radial stress.

σθθ,σΘΘ

Dimension and dimensionless circumferential stress.

υ

Poisson’s ratio of material of the fin.

Ω

Dimensionless exponent of the decayed heat flux.

## References

1. 1.
Takeuti Y, Furukawa T (1981) Some considerations on thermal shock problems in a plate. J Appl Mech 48:113–118
2. 2.
Prevost JH, Tao D (1983) Finite element analysis of dynamic coupled thermoelasticity problems with relaxation time. ASME J Appl Mech 50:817–822
3. 3.
Chen CK, Chen HT (1988) Application of hybrid Laplace transform finite-difference method to transient heat conduction problem. Numer Heat Transf 14:343–356
4. 4.
Yang YC, Chen CK (1986) Thermoelastic transient response of an infinitely long annular cylinder composed of two different materials. Int J Eng Sci 24:569–581
5. 5.
Sherief HH, Anwar MN (1989) A problem in generalized thermoelasticity for an infinitely long annular cylinder composed of two different materials. J Therm Stresses 12:529–543
6. 6.
Wagner P (1994) Fundamental matrix of the system of dynamic linear thermoelasticity. J Therm Stresses 17:549–569
7. 7.
Jane KC, Lee ZY (1999) Thermoelastic transient response of an infinitely long annular multilayered cylinder. Mech Res Commun 26:709–718
8. 8.
Eslami MR, Shakeri M, Ohadi AR, Shiari B (1999) Coupled thermoelasticity of shells of revolution: effect of normal stress and coupling. AIAA J 37:496–504
9. 9.
Tehrani PH, Eslami MR (1998) Two-D time harmonic dynamic coupled thermoelasticity analysis by BEM formulation. Eng Anal Bound Elem 22:245–250
10. 10.
Abd-Alla AM, Abd-Alla AN, Zeidan NA (2000) Thermal stresses in a non-homogeneous orthotropic elastic multilayered cylinder. J Therm Stresses 23:413–428
11. 11.
El-Naggar AM, Abd-Alla AB, Fahmy MA, Ahmed SM (2002) Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Heat Mass Transf 39:41–46
12. 12.
Abd-Alla AM, El-Naggar AM, Fahmy MA (2003) Magneto-thermoelastic problem in non-homogeneous isotropic cylinder. Heat Mass Transf 39:625–629
13. 13.
Lee HL, Yang YC (2001) Inverse problem of coupled thermoelasticity for prediction of heat flux and thermal stresses an annular cylinder. Int Commun Heat Mass Transf 28:661–670
14. 14.
Chen CK, Hung CI, Lee ZY (2001) Transient thermal stresses analysis of multilayered hollow cylinder. Acta Mech 151:75–88
15. 15.
Abd-El-Salam MR, Abd-Alla AM, Hosham Hany A (2007) A numerical solution of magneto-thermoelastic problem in non-homogenous isotropic cylinder by the finite difference method. Appl Math Model 31(8):1662–1670
16. 16.
Jabbari M, Dehbani H, Eslami MR (2010) An exact solution for classic coupled thermoelasticity in spherical coordinates. J Press Vessel Technol 132:1–11
17. 17.
Yang YC, Chu SS (2001) Transient coupled thermoelastic analysis of an annular fin. Int Commun Heat Mass Transf 28:1103–1114
18. 18.
Hung CI, Chen CK, Lee ZY (2001) Thermoelastic transient response of multilayered hollow cylinder with initial interface pressure. J Therm Stresses 24:987–1006
19. 19.
Lee H, Yang Y, Chu S (2002) Transient thermoelastic analysis of an annular fin with coupling effect and variable heat transfer coefficient. J Therm Stresses 25(12):1105–1120
20. 20.
Manoach E, Ribeiro P (2004) Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams. Int J Mech Sci 46:1589–1606
21. 21.
Noda N, Hetnarski RB, Tanigawa Y (2000) Thermal stresses. Lastran, New York Google Scholar
22. 22.
Wachtman JB, Tefft WE, Stinchfied RP (1960) Elastic constants of synthetic crystal corundum at room temperature. J Res Natl Bur Stand, Phys Chem 64A:211–228 Google Scholar
23. 23.
Abd-Alla AM, Mahmoud SR (2010) Magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model. Meccanica 45(4):451–462
24. 24.
Parnell WJ (2006) Coupled thermoelasticity in a composite half-space. J Eng Math 56(1):1–21
25. 25.
Shih W, Chang S, Tsai H (2010) Thermal prediction for transient coupled thermoelastic problem of a two-layer composite material exposed to radiation. J Therm Stresses 33(6):577–594
26. 26.
Bagri A, Eslami MR (2008) Generalized thermoelasticity of functionally graded annular disk considering the Lord-Shulman theory. Compos Struct 83:168–179
27. 27.
Yang YC, Chu SS (2001) Transient coupled thermoelastic analysis of an annular fin. Int Commun Heat Mass Transf 28(8):1103–1114
28. 28.
Bahtui A, Eslami MR (2007) Coupled thermoelasticity of functionally graded cylindrical shells. Mech Res Commun 34:1–18
29. 29.
Bakhshi M, Bagri A, Eslami MR (2006) Coupled thermoelasticity of functionally graded disk. Mech Adv Mat Struct 13:214–225 Google Scholar
30. 30.
Hetnarski RB, Eslami MR (2009) Thermal stresses-advanced theory and applications. Solid mechanics and its applications, vol XXXIV. Springer, New York
31. 31.
Chapra CS (2004) Applied numerical methods with MATLAB for engineering and science. McGraw-Hill, New York Google Scholar

## Authors and Affiliations

• A. M. Abd-Alla
• 1
• 3
• S. R. Mahmoud
• 2
• 3
• S. M. Abo-Dahab
• 1
• 4
1. 1.Math. Dept., Faculty of ScienceTaif UniversityTaifSaudi Arabia
2. 2.Math. Dept., Faculty of ScienceKing Abdul Aziz UniversityJeddahSaudi Arabia
3. 3.Math. Dept., Faculty of ScienceSohagEgypt
4. 4.Math. Dept., Faculty of ScienceSVUQenaEgypt