Abstract
A mathematical model of two-temperature phase-lag Green-Naghdi thermoelasticty theories based on fractional derivative heat transfer is given. The GN theories as well as the theories of coupled and of generalized thermoelasticity with thermal relaxation follow as limit cases. The resulting non dimensional coupled equations together with the Laplace transforms techniques are applied to a specific problem of a half space subjected to arbitrary heating which is taken as a function of time and is traction free. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity with one relaxation time. The effects of temperature discrepancy and fractional order parameters on copper-like material are discussed in different types of GN theories.