Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material with Variable Thermal Conductivity
Youssef Hamdy M. 1, El-Bary A. A. 2
1Mathematical Department, Faculty of Engineering, Umm Al-Qurach University
PO. 5555, Makkah, Saudi Arabia
e-mail: yousefanne@yahoo.com
2Basic and Applied Science Dept. Arab Academy for Science and Technology, Alexandria, Egypt
e-mail: aaelbary@aast.edu
Received:
Rec. 20 September 2005
DOI: 10.12921/cmst.2006.12.02.165-171
OAI: oai:lib.psnc.pl:625
Abstract:
One-dimensional generalized thermoelastic mathematical model with variable thermal conductivity for heat conduction problem is constructed for a layered thin plate. The basic equations are transformed by Laplace transform and solved by a direct method. The solution was applied to a plate of sandwich structure, which is thermally shocked, and traction free in the outer sides. The inverses of Laplace transforms are obtained numerically. The temperature, the stress and the displacement distributions are represented graphically.
Key words:
composite materials, generalized thermoelasticity, thermal shock, thermoelasticity, variable thermal conductivity
References:
[1] H. Lourd and Y. Shulman, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. and Solids 15, 299-309 (1967).
[2] R. Dhaliwal and H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math. 38, 1-8 (1980).
[3] A. Roberts, On the steady motion of a line load over a coupled thermoelastic half-space subsonic case, Quart. J. Mech. Appl. Math. 25, 497-511 (1972).
[4] F. Ziegler, Ebene Wellenausbreitung im Halbraum bei Zufallserregung Spannungs-und Temperaturfeld, Acta Mechanica 2, 307-327 (1966).
[5] Y. Yang and C. Chen, Thermoelastic transient response of an infinitely long annular cylinder composed of two different materials, Int. J. Eng. Sci. 24, 569-581 (1986).
[6] I. Müller, The coldness, a universal function in thermoelastic solids, Arch. Rat. Mech. Anal. 41, 319 (1971).
[7] A. Green and N. Laws, On the entropy production inequality, Arch. Rat. Anal. 54, 7 (1972).
[8] A Green and K. Lindsay, Thermoelasticity, J. Elast. 2, 1 (1972).
[9] E. Şuhubi, Thermoelastic solids, in: A. C. Eringen (ed), Cont. Phys II, Academic Press, New York, Chapter 2 (1987).
[10] S. Erbay and E. Şuhubi, Longitudinal wave propagation in a generalized thermo-elastic cylinder, J. Therm. Stresses 90, 279 (1986).
[11] J. Ignaczak, A strong discontinuity wave in thermoelastic with relaxation times, J. Thermal Stresses 8, 25 (1985).
[12] J. Ignaczak, Decomposition theorem for thermoelasticity with finite wave speeds, J. Thermal Stresses 1, 41 (1978).
[13] D. P. H. Hasselman and R. A. Heller, Thermal Stresses in Sever Environments, Plenum Press, New York (1980).
[14] N. M. El-Maghraby and H. M. Youssef, State Space Approach to Generalized Thermoelastic Problem with Thermo-Mechanical Shock, J. Applied Mathematics and Computation, 156, Issue 2, P. 577-586 (2004).
[15] G. Hanig and U. Hirdes, A method for the numerical inversion of Laplace transform, J. Comp. Appl. Math., 10, 113-132 (1984).
[16] A. A. El-Bary and H. M. Youssef, Thermal Shock Problem for one dimension generalized thermoelastic layered composite material, accepted for publication in Math. and Comp. App. (2005)
[17] S. Minagawa, Nonlinear-eigenvalue problem for heat conduction in a one-dimensional coupled thermoelasticlayered composite, Int. J. Eng. Sci. 25, 1337-1342 (1987).
One-dimensional generalized thermoelastic mathematical model with variable thermal conductivity for heat conduction problem is constructed for a layered thin plate. The basic equations are transformed by Laplace transform and solved by a direct method. The solution was applied to a plate of sandwich structure, which is thermally shocked, and traction free in the outer sides. The inverses of Laplace transforms are obtained numerically. The temperature, the stress and the displacement distributions are represented graphically.
Key words:
composite materials, generalized thermoelasticity, thermal shock, thermoelasticity, variable thermal conductivity
References:
[1] H. Lourd and Y. Shulman, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. and Solids 15, 299-309 (1967).
[2] R. Dhaliwal and H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math. 38, 1-8 (1980).
[3] A. Roberts, On the steady motion of a line load over a coupled thermoelastic half-space subsonic case, Quart. J. Mech. Appl. Math. 25, 497-511 (1972).
[4] F. Ziegler, Ebene Wellenausbreitung im Halbraum bei Zufallserregung Spannungs-und Temperaturfeld, Acta Mechanica 2, 307-327 (1966).
[5] Y. Yang and C. Chen, Thermoelastic transient response of an infinitely long annular cylinder composed of two different materials, Int. J. Eng. Sci. 24, 569-581 (1986).
[6] I. Müller, The coldness, a universal function in thermoelastic solids, Arch. Rat. Mech. Anal. 41, 319 (1971).
[7] A. Green and N. Laws, On the entropy production inequality, Arch. Rat. Anal. 54, 7 (1972).
[8] A Green and K. Lindsay, Thermoelasticity, J. Elast. 2, 1 (1972).
[9] E. Şuhubi, Thermoelastic solids, in: A. C. Eringen (ed), Cont. Phys II, Academic Press, New York, Chapter 2 (1987).
[10] S. Erbay and E. Şuhubi, Longitudinal wave propagation in a generalized thermo-elastic cylinder, J. Therm. Stresses 90, 279 (1986).
[11] J. Ignaczak, A strong discontinuity wave in thermoelastic with relaxation times, J. Thermal Stresses 8, 25 (1985).
[12] J. Ignaczak, Decomposition theorem for thermoelasticity with finite wave speeds, J. Thermal Stresses 1, 41 (1978).
[13] D. P. H. Hasselman and R. A. Heller, Thermal Stresses in Sever Environments, Plenum Press, New York (1980).
[14] N. M. El-Maghraby and H. M. Youssef, State Space Approach to Generalized Thermoelastic Problem with Thermo-Mechanical Shock, J. Applied Mathematics and Computation, 156, Issue 2, P. 577-586 (2004).
[15] G. Hanig and U. Hirdes, A method for the numerical inversion of Laplace transform, J. Comp. Appl. Math., 10, 113-132 (1984).
[16] A. A. El-Bary and H. M. Youssef, Thermal Shock Problem for one dimension generalized thermoelastic layered composite material, accepted for publication in Math. and Comp. App. (2005)
[17] S. Minagawa, Nonlinear-eigenvalue problem for heat conduction in a one-dimensional coupled thermoelasticlayered composite, Int. J. Eng. Sci. 25, 1337-1342 (1987).
