Two-Temperature Generalized Thermopiezoelasticity for One Dimensional Problems – State Space Approach
Youssef Hamdy M. 1, El-Bassiouny Ahmed Habib 2
1Faculty of Engineering, Umm Al-Qura University, PO. Box 5555, Makkah, Saudi Arabia
e-mail: yousefanne@yahoo.com
2King Saud University, Faculty of Science, Department of Mathematics P.O.Box 83, 11942 Al-Kharj, Saudi Arabia
e-mail: essambassiouny@hotmail.com
Received:
Received: 25 April 2007; revised: 8 February 2008; accepted: 21 February 2008; published online: 7 April 2008
DOI: 10.12921/cmst.2008.14.01.55-64
OAI: oai:lib.psnc.pl:646
Abstract:
The theory of two-temperature generalized thermoelasticity, based on the theory of Youssef is used to solve boundary value problems of one dimensional piezoelectric half-space with heating its boundary with different types of heating. The governing equations are solved in the Laplace transform domain by using state-space approach of the modern control theory. The general solution obtained is applied to a specific problems of a half-space subjected to three types of heating; the thermal shock type, the ramp type and the harmonic type. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. The conductive temperature, the dynamical temperature, the stress and the strain distributions are shown graphically with some comparisons
Key words:
generalized thermoelasticity, piezoelectric material, state space, two-temperature
References:
[1] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299-309 (1967).
[2] N. Naotak, R. Hetnarski and Y. Tanigawa, (2003), Thermal Stresses (2nd Edition), Taylor & Francis, New York.
[3] J. Ignaczak and A Note, On Uniqueness in Thermoelasticity With One Relaxation Time, J. Thermal Stresses 5, 257-263 (1982).
[4] Müller, The coldness, A universal function in thermoelastic solids, Arch. Ration. Mech. An. 41, 319-332 (1971).
[5] Ahmed S. El-Karamany and Magdy A. Ezzat, Thermal shock problem in generalized thermo-viscoelasticty under four theories, Int. J. of Eng. Sci. 42(7) 649-671 (2004).
[6] A. E. Green and K. E. Lindsay, Thermoelasticity, J. Elasticity 2, I-7 (1972).
[7] D. Y. Tzou, Macro- to microscale heat transfer: the lagging behavior, Taylor & Francis, Washington (1997) DC.
[8] R. D. Mindlin, On the equations of motion of piezoelectric crystals, in: N. I. Muskilishivili, Problems of continuum Mechanics, 70th Birthday Volume, SIAM, Philadelphia, 282-290 (1961).
[9] R. D. Mindlin, Equations of high frequency vibrations of thermo-piezoelectric plate, Int. J. Solids Struct. 10, 625-637 (1974).
[10] W. Nowacki, Some general theorems of thermo-piezoelectricity, J. Therm. Stress. 1, 171-182 (1978).
[11] W. Nowacki, Foundations of linear piezoelectricity, in: H. Parkus (Ed), Electromagnetic interactions in Elastic Solids, Springer, Wein, Chapter 1. (1979),
[12] D. S. Chandrasekharaiah, A generalized thermoelastic wave propagation in a semi-infinite piezoelectric rod, Acta Mech. 71, 39-49 (1988).
[13] M. C. Majhi, Discontinuities in generalized thermoelastic wave propagation in a semi-infinite piezoelectricrod, J. Tech. Phys. 36, 269-278 (1995).
[14] J. N. Sharma, M. Kumar, Plane harmonic waves in piezo-thermoelastic materials, Indian Eng. Mater. Sci. 7, 434-442 (2000).
[15] E. Bassiouny and A. F. Gahleb, A one dimensional problem in the generalized theory of thermopiezoelasticity, In: G. A. Maugin (Ed), The mechanical behavior of electromagnetic solid continua, Elsevier science publisher B. V. (North Holland), IUTAM-IUPAP, 79-84 (1984).
[16] H. Tianhu, T. Xiaogeneg and S. Yapeng, State space approach to one-dimensional shock problem for a semiinfinite piezoelectric rod, Int. J. Eng. Sci. 40, 1081-1097 (2002).
The theory of two-temperature generalized thermoelasticity, based on the theory of Youssef is used to solve boundary value problems of one dimensional piezoelectric half-space with heating its boundary with different types of heating. The governing equations are solved in the Laplace transform domain by using state-space approach of the modern control theory. The general solution obtained is applied to a specific problems of a half-space subjected to three types of heating; the thermal shock type, the ramp type and the harmonic type. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. The conductive temperature, the dynamical temperature, the stress and the strain distributions are shown graphically with some comparisons
Key words:
generalized thermoelasticity, piezoelectric material, state space, two-temperature
References:
[1] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299-309 (1967).
[2] N. Naotak, R. Hetnarski and Y. Tanigawa, (2003), Thermal Stresses (2nd Edition), Taylor & Francis, New York.
[3] J. Ignaczak and A Note, On Uniqueness in Thermoelasticity With One Relaxation Time, J. Thermal Stresses 5, 257-263 (1982).
[4] Müller, The coldness, A universal function in thermoelastic solids, Arch. Ration. Mech. An. 41, 319-332 (1971).
[5] Ahmed S. El-Karamany and Magdy A. Ezzat, Thermal shock problem in generalized thermo-viscoelasticty under four theories, Int. J. of Eng. Sci. 42(7) 649-671 (2004).
[6] A. E. Green and K. E. Lindsay, Thermoelasticity, J. Elasticity 2, I-7 (1972).
[7] D. Y. Tzou, Macro- to microscale heat transfer: the lagging behavior, Taylor & Francis, Washington (1997) DC.
[8] R. D. Mindlin, On the equations of motion of piezoelectric crystals, in: N. I. Muskilishivili, Problems of continuum Mechanics, 70th Birthday Volume, SIAM, Philadelphia, 282-290 (1961).
[9] R. D. Mindlin, Equations of high frequency vibrations of thermo-piezoelectric plate, Int. J. Solids Struct. 10, 625-637 (1974).
[10] W. Nowacki, Some general theorems of thermo-piezoelectricity, J. Therm. Stress. 1, 171-182 (1978).
[11] W. Nowacki, Foundations of linear piezoelectricity, in: H. Parkus (Ed), Electromagnetic interactions in Elastic Solids, Springer, Wein, Chapter 1. (1979),
[12] D. S. Chandrasekharaiah, A generalized thermoelastic wave propagation in a semi-infinite piezoelectric rod, Acta Mech. 71, 39-49 (1988).
[13] M. C. Majhi, Discontinuities in generalized thermoelastic wave propagation in a semi-infinite piezoelectricrod, J. Tech. Phys. 36, 269-278 (1995).
[14] J. N. Sharma, M. Kumar, Plane harmonic waves in piezo-thermoelastic materials, Indian Eng. Mater. Sci. 7, 434-442 (2000).
[15] E. Bassiouny and A. F. Gahleb, A one dimensional problem in the generalized theory of thermopiezoelasticity, In: G. A. Maugin (Ed), The mechanical behavior of electromagnetic solid continua, Elsevier science publisher B. V. (North Holland), IUTAM-IUPAP, 79-84 (1984).
[16] H. Tianhu, T. Xiaogeneg and S. Yapeng, State space approach to one-dimensional shock problem for a semiinfinite piezoelectric rod, Int. J. Eng. Sci. 40, 1081-1097 (2002).
