Dimensional reduction of a piezoelectric composite rod

In the present paper, a new generalized Timoshenko model is constructed for a composite rod with embedded or attached piezoelectric materials. This model is applicable to composite rods without prescribed electric potential along the lateral surfaces. The Variational-Asymptotic Method (VAM) is applied as a mathematical tool to carry out the dimensional reduction process. The present reduced model captured the effects of dielectric as well as the polarization of the piezoelectric material, which justifies its coupled electromechanical nature. First, the three-dimensional electromechanical enthalpy is asymptotically approximated by VAM using the slenderness of the rod as the small parameter and subsequently an equivalent one-dimensional electromechanical enthalpy is developed. Energy terms, which are asymptotically correct up to the second order are kept in the approximate enthalpy expression. For engineering applications, the approximate enthalpy is then transformed into a generalized Timoshenko model which has the traditional six mechanical degrees of freedom along with an extra one-dimensional electric degree of freedom.