Cross-sectional analysis of nonhomogeneous anisotropic active slender structures

A general formulation for the reduction of the three-dimensional problem of electrothermoelasticity in slender solids to an arbitrarily defined reference line is presented. The dimensional reduction is based on a variational-asymptotic formulation, using the slenderness ratio as small parameter. In the proposed scheme, the coupled linear electroelastic equations are solved at the cross-sectional level using the finite element method. Furthermore, modal components of the displacement field are added to introduce arbitrary deformation shapes into the one-dimensional analysis, and arbitrary electric modes are used to define applied electric fields at the cross section. This results in a general definition of a coupled electroelastic stiffness, which can be used in virtually all composite and active beam formulations, as well as in the development of new low-order high-accuracy reduced models for active structures. Finally, the formulation also yields recovery relations for the elastic and electric fields in the original three-dimensional solid, once the one-dimensional problem is solved. The method has been implemented in a computer program (UM/VABS) and numerical results are presented for active anisotropic beam cross sections of simple geometries, which are shown to compare very well with three-dimensional finite element analysis.