A beam theory consistent with three-dimensional thermo-elasticity

We propose a model for thermo-elastic beams, consistent with the theory of linear three-dimensional thermo-elasticity and deduced by a suitable version of the principle of virtual powers. Dimensional reduction is achieved by postulating convenient a priori representations for mechanical and thermal displacements, the latter playing the role of an additional kinetic variable. Such representations are regarded as internal constraints, some involving the first, others, the second, gradient of deformation and thermal displacements; these constraints are maintained by reactive stresses and hyper- stresses of the type occurring in non-simple elastic materials of grade two, and by reactive entropy influxes and hyper- influxes.