Thermal Stresses in Severe Environments pp 15-30 | Cite as

# Thermoelasticity with Finite Wave Speeds — A Survey

## Abstract

At present there are at least two different generalizations of the classical linear thermoelasticity. The first one proposed by Green and Lindsay (1972) (G-L Theory) involves two relaxation times of a thermoelastic process, while the second theory due to Lord and Shulman (1967) (L-S Theory) admits only one relaxation time. Both theories have been developed in an attempt to eliminate the paradox of an infinite velocity of thermoelastic propagation inherent in the classical case. In this paper a number of general results concerning these theories for a homogeneous isotropic solid are presented. They include (G-L Theory): 1. Domain of Influence Theorem, 2. Decomposition Theorem, 3. Uniqueness Theorem, and 4. Variational Principle. Theorem 1 asserts that in the new theory thermoelastic disturbances produced by the data of bounded support propagate with a finite velocity only. Theorem 2 which is quite similar to a Boggio result in linear elastodynamics shows that a thermoelastic disturbance can be split into two fields each of which propagates with a different finite velocity. Theorem 3 covers uniqueness for a stress-temperature boundary value problem with arbitrary initial tensorial data. Finally, the variational principle gives an alternative description of the theory in terms of a thermoelastic process (S,q), where S and q represent the stress tensor and the heat flux vector, respectively. In the L-S theory uniqueness of a stress-flux initial-boundary value problem is discussed. A number of suggestions concerning those areas of the theory that are critically in need of further investigation are also given.

## Keywords

Variational Principle Uniqueness Theorem Decomposition Theorem Uranium Dioxide Generalize Thermoelasticity## Preview

Unable to display preview. Download preview PDF.

## References

- [1]D. E. Carlson, Linear thermoelasticity, Encyclopedia of Physics, Mechanics of Solids II, vol. 6a/2, Springer, Berlin, 1972.Google Scholar
- [2]W. Nowacki, “Dynamic Problems of Thermoelasticity,” PWN, Warsaw, and Noordhoff, Leyden, 1975.Google Scholar
- [3]V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, “Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity,”North-Holl. Publ. Co., Amsterdam, New York, Oxford, 1979.zbMATHGoogle Scholar
- [4]H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, vol. 15, pp. 299–309, 1967.zbMATHCrossRefGoogle Scholar
- [5]A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity, vol. 2, pp. 1–7, 1972.zbMATHCrossRefGoogle Scholar
- [6]P. H. Francis, Thermo-mechanical effects in elastic wave propagation-a survey, J. Sound and Vibration, vol. 21, no. 2, pp. 181–192, 1972.zbMATHCrossRefGoogle Scholar
- [7]A. Nayfeh and S. Nemat-Nasser, Thermoelastic waves in solids with thermal relaxation, Acta Mechanica, vol. 12, pp. 35–69, 1971.CrossRefGoogle Scholar
- [8]P. Puri, Plane waves in generalized thermoelasticity, Int. J. Engng. Sci., vol. 11, pp. 735–744, 1973. Errata: Int. J. Engng. Sci., vol. 13, pp. 339-340, 1975.zbMATHCrossRefGoogle Scholar
- [9]V. K. Agarwal, On plane waves in generalized thermoelasticity, Acta Mechanica, vol. 31, pp. 185–198, 1979.zbMATHCrossRefGoogle Scholar
- [10]J. D. Achenbach, The influence of heat conduction on propagating stress jumps, J. Mech. and Physics of Solids, vol. 16, pp. 273–282, 1968.CrossRefGoogle Scholar
- [11]J. Ignaczak, Domain of influence theorem in linear thermo-elasticity, Int. J. Engng. Sci., vol. 16, pp. 139–145, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
- [12]J. Ignaczak, Thermoelastic Counterpart to Boggio’s Theorem of Linear Elastodynamics, Bull. Acad. Polon. Sci., Ser. Tech., vol. 24, no. 3, pp. 129–137, 1976.Google Scholar
- [13]J. Ignaczak, Decomposition theorem for thermoelasticity with finite wave speeds, J. Thermal Stresses, vol. 1, no. 1, pp. 41–52, 1978.CrossRefGoogle Scholar
- [14]J. H. Weiner, A uniqueness theorem for the coupled thermo-elastic problem, Q. Appl. Math., vol. 15, pp. 102–105, 1957.MathSciNetzbMATHGoogle Scholar
- [15]J. Ignaczak, A uniqueness theorem for stress-temperature eqs. of dynamic thermoelasticity, J. Thermal Stresses, vol. 1, no. 2, pp. 163–170, 1978.MathSciNetCrossRefGoogle Scholar
- [16]R. E. Nickell and J. L. Sackman, Variational principles for linear coupled thermoelasticity, Q. Appl. Math., vol. 26, pp. 11–26, 1968.MathSciNetzbMATHGoogle Scholar
- [17]J. Ignaczak, Variational characterization of stress and heat flux in dynamic thermoelasticity, in Polish, Matematyka, WSP, Kielce 1979, in print.Google Scholar
- [18]E. B. Popov, Dynamic coupled problem of thermoelasticity for a semi-space taking into account finite speed of heat propagation, in Russian, Prikl, Mat. Mekh., vol. 31, pp. 328–334, 1967.Google Scholar
- [19]J. Ignaczak, Uniqueness in generalized thermoelasticity, J. Thermal Stresses, vol. 2, no. 2, pp. 171–175, 1979.CrossRefGoogle Scholar
- [20]Y. H. Pao and D. K. Banerjee, A theory of anisotropic thermoelasticity at law reference temperature, J. Thermal Stresses, vol. 1, no. 1, pp. 99–112, 1978.CrossRefGoogle Scholar