Cross-Sectional Analysis of Nonhomogeneous Anisotropic Active Slender Structures

Abstract

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 variationalasymptotic 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 onedimensional 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.