In this work, the field equations of the linear theory of thermoelasticity have been constructed in the context of a new consideration of Fourier law of heat conduction with time-fractional order and three-phase lag. A uniqueness and reciprocity theorems are proved. One-dimensional application for a half-space of elastic material in the presence of heat sources has been solved using Laplace transform and state space techniques Ezzat (Canad J Phys Rev 86:1241-1250, 2008. According to the numerical results and its graphs, conclusion about the new theory has been established.Keywords Thermoelasticity · Modified Fourier law · Three-phase lag · Modified Riemann-Liouville fractional derivative · New fractional Taylor's series · State space approach · Fractional calculus Nomenclature λ, μ Lame's constants ρ density C E specific heat at constant strainstrength of the heat source F i mass force q i components of heat flux σ i j components of stress tensor

In this work, a new theory of thermodiffusion in elastic solids is derived using the methodology of fractional calculus. The theories of coupled thermoelastic diffusion and of generalized thermoelastic diffusion problem with one relaxation time follow as limit cases. A variational theorem is then obtained for the governing equations. Finally, a uniqueness and reciprocity theorems for these equations are derived.

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