Boundary Integral Equations Formulation for Fractional Order Thermoelasticity
Tiwari Rakhi *, Mukhopadhyay Santwana
Department of Mathematical Sciences, Indian Institute of Technology
(Banaras Hindu University), Varanasi-221 005, India
*E-mail: rakhibhu2117@gmail.com
Received:
Received: 10 March 2014; revised: 16 May 2014; accepted: 20 May 2014; published online: 24 June 2014
DOI: 10.12921/cmst.2014.20.02.49-58
Abstract:
The present work is concerned with the boundary integral equation formulation for the solutions of equations under fractional order thermo elasticity in a three dimensional Euclidean space. A mixed initial-boundary value problem is considered and the fundamental solutions of the corresponding coupled differential equations are obtained in the Laplace transform domain. We employ one reciprocal relation in the present context and formulate the boundary integral equations on the basis of our fundamental solutions.Then the formulation is illustrated with a suitable example.
Key words:
boundary integral equation method, fractional order thermoelasticity, fundamental solutions, thermoelasticity
References:
[1] C.A. Brebbia, The boundary element method for engineers, Pentech Press, London 1978.
[2] C.A. Brebbia and S. Walker, Boundary Element Method techniques in Engineering, Newness-Butterworths, London 1980.
[3] R.T. Fenner, Boundary element for stress problems. Seminar on finite elements or boundary elements, 15th June 1982.
[4] M.A. Jawson, Integral equation methods in potential theory I, Proc. R. Soc. Lond. A 275, 23-32 (1963).
[5] G.T. Symm, Integral equation methods in potential theory II, Proc. R. Soc. Lond. A 275, 33-46 (1963).
[6] F.J. Rizzo, and D.J. Shippi, An advanced Boundary integral equation method for three-dimensional thermoelasticity, Int. J. Numer.
Methods. Eng. 11, 1753-1763 (1977).
[7] J. Chen, and G.F. Dargush, Boundary element method for dynamic poroelastic and thermoelastic analyses, Int. J. Solids. Struct. 32,
2257-2278 (1995).
[8] T.A. Cruse and F.J. Rizzo, Boundary integral equation methods-computational applications, ASME Boundary Integral Equation
Meth. Conf. Proc 118-123 (1985).
[9] P.K. Banerjee, R. Butterfield, Boundary element methods in engineering science, McGraw-Hill, London 1981.
[10] C.A. Brebbia, J.C.F. Tells and L.C. Wrobel, Boundary element techniques: theory and applications in engineering, Springer, Hei-
delberg 1984.
[11] F. Ziegler and H. Irschik, Thermal stress analysis based on Maysel’s formula, in: Hetnarski, RB (ed.) Mechanical and Mathematical
methods, Thermal stresses II, North Holland, Amsterdam 1987.
[12] M. Tanaka, T. Matsumoto and M. Moradi, Application of boundary element method to 3-D problems of coupled thermoelasticity,
Eng. Anal. Boundary. Elem. 16, 297-303 (1995).
[13] M. Anwar and H. Sherief, Boundary integral equation formulation of generalized thermoelasticity in a Laplace transform domain,
Appl. Math. Model. 12,161-166 (1988).
[14] M. Kogl and L. Gaul, A boundary element method for anisotropic coupled thermoelasticity, Arch. Appl. Math. 73, 377-398 (2003).
[15] A.S. El-Karamany and M.A. Ezzat, Analytical aspects in boundary integral equation formulation for generalized linear micropolar
thermoelasticity, Int. J. Mech. Sci. 46, 389-409 (2004).
[16] A.S. El-Karamany and M.A. Ezzat, Boundary Integral equation formulation for the generalized thermoviscoelasticity with two
relaxation times, Appl. Math. Comput. 151,347-362 (2004).
[17] A.S. El-Karamany, Boundary integral equation formulation for generalized micro polar thermoviscoelasticity, Int. J. Eng. Sci. 42,
157-186 (2004).
[18] R. Prasad, S. Das, S. Mukhopadhyay, Boundary integral equation for coupled thermoelasticity with three phage lags, Mathematics
and Mechanics of Solids 18(1), 44-58 (2013).
[19] S. Semwal, S. Mukhopadhyay, Boundary integral equation formulation for generalized thermo elastic diffusion – Analytical Asects,
Appl. Math. Model. (2014).
[20] C. Remillat, M.R. Hassan, F. Scarpa, Small Amplitude Dynamic Properties of Ni48Ti46Cu6 SMA Ribbons: Experimental Results
and Modelling, ASME Journal of Engineering Materials and Technology 128(3), 260-267 (2007).
[21] I. Podlubny, Fractional differential equations. An introduction to fractional order derivatives, Fractional differential equations,
some methods of their solutions and some of their applications. London-Tokyo-Toronto, Academic Press Volume 198, (1999).
[22] D.Y. Tzou, Macro- to Microscale Heat Transfer, Taylor and Francis, Washington, DC. (1997).
[23] D.S. Chandrasekharaiah , Heat flux dependent micropolar thermoelasticity. International Journal of Engineering Science, 24, 1389-
1395 (1986).
[24] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51, 705-729 (1998).
[25] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compt. Rend. 247,
431-433 (1958).
[26] P. Vernotte, Les paradoxes de la theorie continue de l‘equation de la chaleur, Compte Rendus. 246, 3154-3155 (1958).
[27] P. Vernotte, Some possible complications in the phenomena of thermal conduction, Compt Rend. 252, 2190-2191 (1961).
[28] H.H. Sherief, A. El Sayed and A. El-Latief, Fractional Order Theory of Thermoelasticity, Int. J. Solids Struct. 47, 269-275 (2010).
The present work is concerned with the boundary integral equation formulation for the solutions of equations under fractional order thermo elasticity in a three dimensional Euclidean space. A mixed initial-boundary value problem is considered and the fundamental solutions of the corresponding coupled differential equations are obtained in the Laplace transform domain. We employ one reciprocal relation in the present context and formulate the boundary integral equations on the basis of our fundamental solutions.Then the formulation is illustrated with a suitable example.
Key words:
boundary integral equation method, fractional order thermoelasticity, fundamental solutions, thermoelasticity
References:
[1] C.A. Brebbia, The boundary element method for engineers, Pentech Press, London 1978.
[2] C.A. Brebbia and S. Walker, Boundary Element Method techniques in Engineering, Newness-Butterworths, London 1980.
[3] R.T. Fenner, Boundary element for stress problems. Seminar on finite elements or boundary elements, 15th June 1982.
[4] M.A. Jawson, Integral equation methods in potential theory I, Proc. R. Soc. Lond. A 275, 23-32 (1963).
[5] G.T. Symm, Integral equation methods in potential theory II, Proc. R. Soc. Lond. A 275, 33-46 (1963).
[6] F.J. Rizzo, and D.J. Shippi, An advanced Boundary integral equation method for three-dimensional thermoelasticity, Int. J. Numer.
Methods. Eng. 11, 1753-1763 (1977).
[7] J. Chen, and G.F. Dargush, Boundary element method for dynamic poroelastic and thermoelastic analyses, Int. J. Solids. Struct. 32,
2257-2278 (1995).
[8] T.A. Cruse and F.J. Rizzo, Boundary integral equation methods-computational applications, ASME Boundary Integral Equation
Meth. Conf. Proc 118-123 (1985).
[9] P.K. Banerjee, R. Butterfield, Boundary element methods in engineering science, McGraw-Hill, London 1981.
[10] C.A. Brebbia, J.C.F. Tells and L.C. Wrobel, Boundary element techniques: theory and applications in engineering, Springer, Hei-
delberg 1984.
[11] F. Ziegler and H. Irschik, Thermal stress analysis based on Maysel’s formula, in: Hetnarski, RB (ed.) Mechanical and Mathematical
methods, Thermal stresses II, North Holland, Amsterdam 1987.
[12] M. Tanaka, T. Matsumoto and M. Moradi, Application of boundary element method to 3-D problems of coupled thermoelasticity,
Eng. Anal. Boundary. Elem. 16, 297-303 (1995).
[13] M. Anwar and H. Sherief, Boundary integral equation formulation of generalized thermoelasticity in a Laplace transform domain,
Appl. Math. Model. 12,161-166 (1988).
[14] M. Kogl and L. Gaul, A boundary element method for anisotropic coupled thermoelasticity, Arch. Appl. Math. 73, 377-398 (2003).
[15] A.S. El-Karamany and M.A. Ezzat, Analytical aspects in boundary integral equation formulation for generalized linear micropolar
thermoelasticity, Int. J. Mech. Sci. 46, 389-409 (2004).
[16] A.S. El-Karamany and M.A. Ezzat, Boundary Integral equation formulation for the generalized thermoviscoelasticity with two
relaxation times, Appl. Math. Comput. 151,347-362 (2004).
[17] A.S. El-Karamany, Boundary integral equation formulation for generalized micro polar thermoviscoelasticity, Int. J. Eng. Sci. 42,
157-186 (2004).
[18] R. Prasad, S. Das, S. Mukhopadhyay, Boundary integral equation for coupled thermoelasticity with three phage lags, Mathematics
and Mechanics of Solids 18(1), 44-58 (2013).
[19] S. Semwal, S. Mukhopadhyay, Boundary integral equation formulation for generalized thermo elastic diffusion – Analytical Asects,
Appl. Math. Model. (2014).
[20] C. Remillat, M.R. Hassan, F. Scarpa, Small Amplitude Dynamic Properties of Ni48Ti46Cu6 SMA Ribbons: Experimental Results
and Modelling, ASME Journal of Engineering Materials and Technology 128(3), 260-267 (2007).
[21] I. Podlubny, Fractional differential equations. An introduction to fractional order derivatives, Fractional differential equations,
some methods of their solutions and some of their applications. London-Tokyo-Toronto, Academic Press Volume 198, (1999).
[22] D.Y. Tzou, Macro- to Microscale Heat Transfer, Taylor and Francis, Washington, DC. (1997).
[23] D.S. Chandrasekharaiah , Heat flux dependent micropolar thermoelasticity. International Journal of Engineering Science, 24, 1389-
1395 (1986).
[24] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51, 705-729 (1998).
[25] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compt. Rend. 247,
431-433 (1958).
[26] P. Vernotte, Les paradoxes de la theorie continue de l‘equation de la chaleur, Compte Rendus. 246, 3154-3155 (1958).
[27] P. Vernotte, Some possible complications in the phenomena of thermal conduction, Compt Rend. 252, 2190-2191 (1961).
[28] H.H. Sherief, A. El Sayed and A. El-Latief, Fractional Order Theory of Thermoelasticity, Int. J. Solids Struct. 47, 269-275 (2010).
