A coupled Timoshenko model for smart slender structures

In this paper, a generalized Timoshenko model has been developed for prismatic, beam-like slender structures with embedded or surface mounted piezoelectric type smart materials. Starting from a geometrically exact formulation of the original, three-dimensional electromechanical problem, we apply the variational asymptotic method to carry out a systematic dimensional reduction. In the process, the three-dimensional electromechanical enthalpy functional is approximated asymptotically using the slenderness as the small parameter to find out an equivalent one-dimensional electromechanical enthalpy functional. For Timoshenko-like refinement over the Euler–Bernoulli beam model, terms up to the second order of the slenderness are kept in the enthalpy expression. As an unified analysis tool, the present model can analyze embedded or surface mounted active layer with arbitrary cross-sectional geometry as two cases of a general one, no special assumptions or modifications need to be made for these two separate types of active inclusions.