Guided by the three-dimensional theory of coupled thermoelasticity with second sound, a system of shear-deformable shell equations is consistently derived in invariant, differential and variational forms for the high-frequency vibrations of temperature-dependent materials. The first part of the paper is concerned with a unified variational principle describing the fundamental equations of thermoelasticity. The differential type of Variational principle is presented by expressing Hamilton's principle for the thermal part of a thermoelastic region and then combining it with its mechanical part. In the second part, the hierarchic system of non-isothermal shell equations is systematically established by use of the variational principle together with Mindlin's kinematic hypothesis for shells. The system of two-dimensional approximate equations which may take account of all the significant mechanical and thermal effects, including the temperature dependency of material, governs the extensional, thickness-shear, flexural and torsional as well as coupled vibrations of shells of uniform thickness. Lastly, in the third part, emphasis is placed on certain cases involving special material, geometry and kinematics. Besides, a theorem is devised so as to enumerate the initial and boundary conditions sufficient for the uniqueness in solutions of the system of non-isothermal shell equations. (C) 2001 Elsevier Science Ltd. All rights reserved.