Mathematical model of generalized thermoelastic infinite medium with a spherical cavity and fractional order strain
Eman A. N. Al-Lehaibi
- Mathematics Department, College of Science and Arts-Sharoura, Najran University, KSA.
In this paper, a new mathematical model of a thermoelastic isotropic unbounded medium contains a spherical cavity thermally shocked under generalized thermo-elasticity with the fractional order strain model. The governing system of the partial differential equations has been derived in Laplace transform domain, and the inversion was done numerically by using the sum of Riemann approximation techniques. The numerical outputs of the displacement, the temperature, the stress, and the strain have been obtained and presented graphically. The fractional order parameter has an essential consequence on the stress, the strain, and the displacement distributions while its effect on the temperature increment distribution is very limited.
- Generalized thermo-elasticity
- spherical cavity
- fractional calculus
- fractional strain
N. S. Al-Huniti, M. Al-Nimr, M. Naji , Dynamic response of a rod due to a moving heat source under the hyperbolic heat conduction model , J. Sound Vib., 242 (2001), 629–640.
M. A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys., 27 (1956), 240–253.
S. Erbay, E. S. Suhubi , Longitudinal wave propagation in a generalized thermoelastic cylinder, J. Therm. Stresses, 9 (1986), 279–295.
T. Furukawa, N. Noda, F. Ashida , Generalized thermoelasticity for an infinite body with a circular cylindrical hole, JSME Int. J. I-Solid M., 33 (1990), 26–32.
A. E. Green, N. Laws, On the entropy production inequality, Arch. Rational Mech. Anal., 45 (1972), 47–53.
A. E. Green, K. A. Lindsay , Thermoelasticity, J. Elasticity, 2 (1972), 1–7.
H. W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15 (1967), 299–309.
R. L. Magin, T. J. Royston , Fractional-order elastic models of cartilage: A multi-scale approach, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 657–664.
J. C. Misra, N. C. Chattopadhyay, S. C. Samanta , Thermoviscoelastic waves in an infinite aeolotropic body with a cylindrical cavity–a study under the review of generalised theory of thermoelasticity, Computers & structures, 52 (1994), 705–717.
J. C. Misra, S. C. Samanta, A. K. Chakrabarti, S. C. Misra, Magnetothermoelastic interaction in an infinite elastic continuum with a cylindrical hole subjected to ramp-type heating, Int. J. Eng. Sci., 29 (1991), 1505–1514.
I. Mller , The Coldness, a Universal Function in Thermo-Elastic Solids, Arch. Rat. Mech. Anal., 41 (1971), 319–332.
J. L. Schiff , The Laplace transform: theory and applications, Springer Science & Business Media, New York (2013)
H. H. Sherief, M. N. Anwar, Two-dimensional generalized thermoelasticity problem for an infinitely long cylinder, J. Thermal Stresses, 17 (1994), 213–227.
E. S. Suhubi , Thermoelastic solids: Continuum Mechanics of Single-Substance Bodies, Elsevier, 1975 (1975), 173–265.
D. Y. Tzou, Macro-to microscale heat transfer: the lagging behavior, John Wiley & Sons, U.S.A. (2014)
H. M. Youssef , Dependence of modulus of elasticity and thermal conductivity on reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity , Appl. Math. Mech., 26 (2005), 470–475.
H. M. Youssef , State-space approach on generalized thermoelasticity for an infinite material with a spherical cavity and variable thermal conductivity subjected to ramp-type heating , Can. Appl. Math. Q., 13 (2005), 369–390.
H. M. Youssef , Theory of generalized thermoelasticity with fractional order strain , J. Vib. Control, 22 (2016), 3840–3857.