Wave Propagation and Fundamental Solution of Initially Stressed Thermoelastic Diffusion with Voids

Subject Areas : Engineering

R Kumar ¹ (Department of Mathematics, Kurukshetra University)
R. Kumar ² (Department of Mathematics, Kurukshetra University)

Received: 2011-09-10 Accepted : 2011-10-27 Published : 2011-09-30

Keywords: Initial stress, Plane waves, Fundamental solution, Thermoelastic diffusion with voids, Steady oscillations,

Abstract :

The present article deals with the study of propagation of plane waves in isotropic generalized thermoelastic diffusion with voids under initial stress. It is found that, for two dimensional model of isotropic generalized thermoelastic diffusion with voids under initial stress, there exists four coupled waves namely, P wave, Mass Diffusion (MD) wave, thermal (T) wave and Volume Fraction (VF) wave. The phase propagation velocities and attenuation quality factor of these plane waves are also computed and depicted graphically. In addition, the fundamental solution of system of differential equations in the theory of initially stressed thermoelastic diffusion with voids in case of steady oscillations in terms of elementary functions has been constructed. Some basic properties of the fundamental solution are established and some particular cases are also discussed.

References:

[1] Biot M.A., 1956, Theory of propagation of elastic waves in a fluid saturated porous solid. I low frequency range, The Journal of the Acoustical Society of America 28: 335-354.

[2] Biot M.A., Willis D.G., 1957, Elastic coefficients of the theory of consolidation, The Journal of the Acoustical Society of America 24: 594-601.

[3] Goodman M.A., Cowin S.C., 1971, A continuum theory of granular material, Archive for Rational Mechanics and Analysis 44: 249-266.

[4] Nunziato J.W., Cowin S.C., 1979, A non-linear theory of elastic materials with voids, Archive for Rational Mechanics and Analysis 72: 175-201.

[5] Cowin S.C., Nunziato J.W., 1983, Linear elastic materials with voids, Journal of Elasticity 13: 125-147.

[6] Iesan D., 1986, A theory of thermoelastic materials with voids, Acta Mechanica 60: 67-89.

[7] Iesan D., 2004, Thermoelastic Models of Continua, Springer, Berlin.

[8] Capriz G., 1989, Continua with microstructure. In: Springer Tracts in Natural Philosophy, edited by C.A. Truesdell 35. Springer, Berlin.

[9] Cowin S.C., 1985, The viscoelastic behavior of linear elastic materials with voids, Journal of Elasticity 15: 185-191.

[10] Iesan D., 1987, A theory of initially stressed thermoelastic material with voids, An. St. Univ. Iasi, S. I-a Matematica, 33: 167-184.

[11] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299-309.

[12] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.

[13] Dhaliwal R.S. Sherief H., 1980, Generalized thermoelasticity for anisotropic media, Quarterly of Applied Mathematics 33: 1-8.

[14] Nowacki W., 1974a, Dynamical problems of thermodiffusion in solids-I, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 55-64.

[15] Nowacki W., 1974b, Dynamical problems of thermodiffusion in solids-II, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 129-135.

[16] Nowacki W., 1975c, Dynamical problems of thermodiffusion in solids-III, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 275-276.

[17] Nowacki W., 1976, Dynamical problems of diffusion in solids, Engineering Fracture Mechanics 8: 261-266.

[18] Sherief H., Saleh, H., 2005, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42: 4484-4493.

[19] Singh B., 2005, Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion, Journal of Earth and System and Sciences 114(2): 159-168.

[20] Singh B., 2006, Reflection of SV waves from free surface of an elastic solid in generalized thermodiffusion, Journal of Sound and Vibration 291(3-5): 764-778.

[21] Aouadi M., 2006, Variable electrical and thermal conductivity in the theory of generalized thermodiffusion, Zeitschrift für angewandte Mathematik und Physik (ZAMP) 57(2): 350-366.

[22] Aouadi M., 2006, A generalized thermoelastic diffusion problem for an infinitely long solid cylinder, International Journal of Mathematics and Mathematical Sciences, Article ID 25976:1-15.

[23] Aouadi M., 2007, A problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 44: 5711-5722.

[24] Gawinecki J. A., Szymaniec A., 2002, Global solution of the cauchy problem in nonlinear thermoelastic diffusion in solid body, Proceedings in Applied Mathematics and Mechanics (PAMM) 1: 446-447.

[25] Gawinecki J.A., Kacprzyk P., Bar-Yoseph P., 2000, Initial boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermoelastic diffusion in solid body, Journal for Analysis and its Applications 19: 121-130.

[26] Sherief H.H., Saleh H., Hamza F., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591-608.

[27] Aouadi M., 2007, Uniqueness and reciprocity theorems in the theory of generalized thermoelastic diffusion, Journal of Thermal Stresses 30: 665-678.

[28] Aouadi M., 2008, Generalized theory of thermoelastic diffusion for anisotropic media, Journal of Thermal Stresses 31: 270-285.

[29] Aouadi M., 2010, A theory of thermoelastic diffusion materials with voids, Zeitschrift für angewandte Mathematik und Physik (ZAMP) 61: 357-379.

[30] Biot M.A., 1965, Mechanics of Incremental Deformation, John Wiley and Sons, New York.

[31] Hetnarski R.B., 1964, The fundamental solution of the coupled thermoelastic problem for small times, Archiwwn Mechhaniki Stosowwanej 16: 23-31.

[32] Hetnarski R.B., 1964, Solution of the coupled problem of thermoelasticity in form of a series of functions, Archiwwn Mechhaniki Stosowwanej 16: 919-941.

[33] Iesan D., 1998, On the theory of thermoelasticity without energy dissipation, Journal of Thermal Stresses 21: 295-307.

[34] Svanadze M., 1988, The fundamental matrix of the linearlized equations of the theory of elastic mixtures, Proceeding I. Vekua Institute of Applied Mathematics, Tbilisi State University 23:133-148.

[35] Svanadze M., 1996, The fundamental solution of the oscillation equations of thermoelasticity theory of mixtures of two solids, Journal of Thermal Stresses 19:633-648.

[36] Svanadze M., 2004, Fundamental solutions of the equations of the theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses 27:151-170.

[37] Svanadze M., Fundamental solution of the system of equations of steady oscillations in the theory of microstrecth, International Journal of Engineering Science 42: 1897-1910.

[38] Svanadze M., 2007, Fundamental solution in the theory of micropolar thermoelasticity for materials with voids, Journal of Thermal Stresses 30: 219-238.

[39] Hormander L., 1983, The analysis of linear partial differential Operators II: Differential operators with constant coefficients, Springer-Verlang, Berlin.

[40] Hormander L., 1963, Linear Partial Differential Operators, Springer-Verlang, Berlin.

[41] Magana A., Quintanilla R., 2006, On the exponential decay of solutions in one dimensional generalized porous-thermo-elasticity, Asymptotic Analysis 49: 173-187.

[42] Aouadi M., 2012, Stability in thermoelastic diffusion theory with voids, Applicable Analysis 91: 121-139.

[43] Sturnin D.V., 2001, On characteristics times in generalized thermoelasticity, Journal of Applied Mechanics, 68: 816-817.

[44] Sharma M.D., 2008, Wave propagation in thermoelastic saturated porous medium, Journal of Earth System Science, 117(6): 951-958.