A novel reduced model for a linearized anisotropic rod with doubly symmetric a cross-section: I. Theory

Abstract

A novel reduced model is constructed for a linearized anisotropic rod with doubly symmetric cross-section. The derivation starts from the Taylor expansion of the displacement vector and the stress tensor. The goal is to establish rod equations for the leading order displacement and the twist angle of the mean line of the rod in an asymptotically consistent way. Fifteen vector differential equations are derived from the 3D (three-dimensional) governing system, and elaborate manipulations between these equations (including the Fourier series expansion of the lateral traction condition) lead to four scalar rod equations: two bending equations, one twisting equation, and one stretching equation. Also, recursive relations are established between the higher order coefficients and the lower order ones, which eliminate most of the unknowns. Six boundary conditions at each edge are obtained from the 3D virtual work principle, and 1D (one-dimensional) virtual work principle is also developed. The rod model has three features: it adopts no ad hoc assumptions for the displacement form and the scalings of the external loadings; it incorporates the bending, twisting, and stretching effects in one uniform framework; and it satisfies the 3D governing system in a point-wise manner.