Dynamic Coupled Thermo-Viscoelasticity of a Spherical Hollow Domain

Document Type: Research Paper


1 M.Sc. Student, Mechanical Engineering Department,South Tehran Branch, Islamic Azad University, Tehran, Iran

2 Amirkabir University of Technology


The generalized coupled thermo-viscoelasticity of hollow sphere subjected to thermal symmetric shock load is presented in this paper. To overcome the infinite speed of thermal wave propagation, the Lord-Shulman theory is considered. Two coupled equations, namely, the radial equation of motion and the energy equation of a hollow sphere are obtained in dimensionless form. Resulting equations are transformed into the Laplace domain using the Laplace transformation and discretized by means of the finite element method along the radius. Week's method is employed to obtain the unknowns in the time domain. The numerical results are provided to examine the propagation of thermal and mechanical waves. The results presented visually as temporal, radial and stress-strain plots to shown wave front of various field variables.


Main Subjects

[1]   Lord, H. W., and Shulman, Y., "A Generalized Dynamical Theory of Thermoelasticity", J. Mech. Phys. Solids, Vol. 15, pp. 299-309, (1967).


[2]   Green, A. E., and Lindsay, K. A., "Thermoelasticity", J. Elast., Vol. 2, pp. 1-7, (1972).


[3]   Green, A. E., and Naghdi, P. M., "A Re-Examination of the Basic Postulates of Thermomechanics", Proc. R. Soc. A, Vol. 432, pp. 171-194, (1991).


[4]   Green, A. E., and Naghdi, P. M., "Thermoelasticity without Energy Dissipation", J. Elast., Vol. 31, pp. 189-208, (1993).


[5]   Müller, I., "The Coldness, a Universal Function in Thermoelastic Bodies", Arch. Ration. Mech. Anal., Vol. 41, pp. 319-332, (1971).


[6]   Green, A. E., and Laws, N., "On the Entropy Production Inequality", Arch. Ration. Mech. Anal., Vol. 45, pp. 47-53, (1972).


[7]   Bagri, A., and Eslami, M. R., "A Unified Generalized Thermoelasticity; Solution for Cylinders and Spheres", Int. J. Mech. Sci., Vol. 49, pp. 1325-1335, (2007).


[8]   Ignaczak, J., "A Note on Uniqueness in Thermoelasticity with One Relaxation Time", J. Therm. Stresses, Vol. 5, pp. 257-263, (1982).


[9]   Chandrasekharaiah, D. S., "Thermoelasticity with Second Sound: A Review", Appl. Mech. Rev., Vol. 39, pp. 355-376, (1986).


[10] Chandrasekharaiah, D. S., "Hyperbolic Thermoelasticity: A Review of Recent Literature", Appl. Mech. Rev., Vol. 51, pp. 705-729, (1998).


[11] Bagri, A., and Eslami, M. R., "Generalized Coupled Thermoelasticity of Functionally Graded Annular Disk Considering the Lord-Shulman Theory", Compos. Struct., Vol. 83, pp. 168-179, (2008).


[12] Jabbari, M., and Dehbani, H., Exact Solution for Lord-Shulman Generalized Coupled Thermoporoelasticity in Spherical Coordinates", in R. B. Hetnarski (Editor), Encyclopedia of Thermal Stresses, Vol. 2, pp. 1412-1426, Springer, Dordrecht, (2014).


[13] Kiani, Y., and Eslami, M. R., "Generalized Thermoelasticity of Rotating Disk", Int. Mech. Aerosp. Eng., Vol. 2, pp. 23-29, (2016).


[14]   Ezzat, M. A., El-Karamany, A. S., and Samaan, A. A., "State-space Formulation to Generalized Thermoviscoelasticity with Thermal Relaxation", J. Therm. Stresses, Vol. 24, pp. 823-846, (2001).


[15]   Othman, M. I. A., Ezzat, M. A., Zaki, S. A., and El-Karamany, A. S., "Generalized Thermo-viscoelastic Plane Waves with Two Relaxation Times", Int. J. Eng. Sci., Vol. 40, pp. 1329-1347, (2002).

[16]   El-Karamany, A. S., and Ezzat, M. A., "On the Boundary Integral Formulation of Thermo-viscoelasticity Theory" Int. J. Eng. Sci., Vol. 40, pp. 1943-1956, (2002).

[17]   Kar, A., and Kanoria, M., "Generalized Thermo-visco-elastic Problem of a Spherical Shell with Three-phase-lag Effect", Appl. Math. Modell. Vol. 33, pp. 3287-3298, (2009).

[18]   Weeks, W. T., "Numerical Inversion of Laplace Transforms using Laguerre Functions", J. ACM, Vol. 13, pp. 419-426, (1966).


[19]   Kanoria, M., and Mallik, S. H., "Generalized Thermoviscoelastic Interaction Due to Periodically Varying Heat Source with Three-phase-lag Effect", Eur. J. Mech. A-Solid, Vol. 29, pp. 695-703, (2010).


[20]   Hetnarski, R. B., and Eslami, M. R., "Thermal Stresses, Advanced Theory and Applications", Springer, Amsterdam, (2009).


[21]   Eslami, M. R., "Finite Elements Methods in Mechanics", Springer, Cham, (2014).


[22]   Wang, Q., and Zhan, H., "On Different Numerical Inverse Laplace Methods for Solute Transport Problems", Adv. Water Resour. Vol. 75, pp. 80-92, (2015).


[23]   Eslami, M. R., and Vahedi, H., "A Galerkin Finite Element Formulation of Dynamic Thermoelasticity for Spherical Problems", Proc. 1989 ASME PVP Conf., Hawaii, (1989).


[24]   Eslami, M. R., and Vahedi, H., "Galerkin Finite Element Displacement Formulation of Coupled Thermoelasticity Spherical Problems", J. Pressure Vessel Technol., Vol. 114, pp. 380-384, (1992).