Mechanics Research Communications 37 (2010) 448–452 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom Fractional order generalized thermoelastic half-space subjected to ramp-type heating Hamdy M. Youssef a,∗ , Eman A. Al-Lehaibi b a Mathematical Department, Faculty of Education, Alexandria University, Egypt b Faculty of Applied Sciences, Mathematical Department, Umm Al-Qura University, Saudi Arabia article info Article history: Received 29 March 2010 Received in revised form 22 May 2010 Available online 9 June 2010 Keywords: Thermoelasticity Generalized thermoelasticity Fractional order State-space abstract In this work, we will construct a mathematical model of an elastic material with constant parameters fills the half-space and the governing equations will be taken into the context of the fractional order generalized thermoelasticity theory (Youssef, 2010). The medium is assumed initially quiescent and Laplace transforms and state-space techniques will be used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a medium subjected to ramp-type heating and traction free. The inverse of the Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effects of the fractional order parameter on all the studied felids. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Because of the advancement of pulsed lasers, fast burst nuclear reactors and particle accelerators, etc. which can supply heat pulses with a very fast time-rise Bargmann (1974), Anisimov et al. (1974), Boley (1980), Qiu and Tien (1993), Tzou (1997), Warren and Chen (1973), Naotak et al. (2003); generalized thermoelastic- ity theory is receiving serious attention of different researchers. The development of the second sound effect has been reviewed by Chandrasekhariah (1986). Now, mainly two different models of generalized thermoelasticity are being extensively used-one pro- posed by Lord and Shulman (1967) and the other proposed by Green and Lindsay (1972). The L–S theory suggests one relaxation time and according to this theory, only Fourier’s heat conduction equation is modified; while G–L theory suggests two relaxation times and both the energy equation and the equation of motion are modified. A method for solving coupled thermoelastic prob- lems by using the state-space approach was developed by Bahar and Hetnarski (1978). Erbay and Suhubi (1986) studied longitudinal wave propagation in an infinite circular cylinder, which is assumed to be made of the generalized thermoelastic material, and thereby obtained the dispersion relation when the surface temperature of the cylinder was kept constant. Generalized thermoelasticity prob- lems for an infinite body with a circular cylindrical hole and for ∗ Corresponding author. Present address: Faculty of Engineering, Umm Al-Qura University, P.O. 5555, Makkah, Saudi Arabia. Tel.: +966 509274732. E-mail address: yousefanne@yahoo.com (H.M. Youssef). an infinite solid cylinder were solved respectively by Furukawa et al. (1990). Chandrasekaraiah and Murthy (1993) studied ther- moelastic interactions in an isotropic homogeneous unbounded linear thermoelastic body with a spherical cavity, in which the field equations were taken in unified forms covering the coupled, L–S and G–L models of thermoelasticity. The effects of mechanical and thermal relaxations in a heated viscoelastic medium containing a cylindrical hole were studied by Misra et al. (1987). Investigations concerning interactions between magnetic and thermal fields in deformable bodies were carried out by Maugin (1988) as well as by Eringen and Maugin (1989). Subsequently Abd-Alla and Maugin (1990) conducted a generalized theoretical study by considering the mechanical, thermal and magnetic field in centro-symmetric magnetizable elastic solids. Recently, in the non classical thermoelasticity theories, Fourier law of heat conduction is replaced by more general equation which includes a relaxation time parameter. The first well-known gener- alized of such a type of Lord and Shulman (1967) which leads to the hyperbolic differential equation of heat conduction. Now, a new formula of heat conduction has been considered in the context of the fractional integral operator definition by Youssef (2010) which is called, the fractional order generalized thermoelas- ticity theory. In the context of this consideration, the heat flux and the entropy increment equations have not any change where the second law of the thermo dynamics doesn’t depend on the modi- fication of the Fourier law of heat conduction. Youssef (2010) has approved the uniqueness of the solutions by this theorem. In this paper, we will construct a model of an elastic material with constant parameters and fills the half-space. The governing 0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.06.003