On the phase- lag Green–Naghdi thermoelasticity theories

• A single –phase - lag of type III, a dual-phase–lag of type II and of type III are proposed.
• A unified heat conduction law and heat transport equation are given, which consolidate the three theories.
• The Coupled theory, The Lord–Shulman theory and the Green–Naghdi theory of type II result from the unified law.
• Dissipative inequality is derived, constitutive equations and thermodynamic restrictions are given.
• Uniqueness theorem and Variational principle are proved. An application is given for isotropic solid.

In this work, we proposed three models of generalized thermoelasticity: a single –phase - lag Green–Naghdi theory of type III, a dual-phase–lag Green–Naghdi theory of type II and of type III. The solid is assumed to be linear anisotropic inhomogeneous. A unified heat conduction law and heat transport equation are given, which consolidate the three theories and also the Lord–Shulman theory and the Green–Naghdi theory of type II. Dissipative inequality is derived, constitutive equations and thermodynamic restrictions are given. The speed of thermal wave propagation is given, for each of the three theories in case of isotropic solid. A uniqueness theorem is proved for the considered theories. Variational characterization of solution is established. An application is given for isotropic thermoelastic solid and the results are presented graphically.