On a consistent rod theory for a linearized anisotropic elastic material: I. Asymptotic reduction method

Abstract

An asymptotic reduction method is introduced to construct a rod theory for a linearized general anisotropic elastic material for space deformation. The starting point is Taylor expansions about the central line in rectangular coordinates, and the goal is to eliminate the two cross-section spatial variables in order to obtain a closed system for displacement coefficients. This is first achieved, in an ‘asymptotically inconsistent’ way, by deducing the relations between stress coefficients from a Fourier series for the lateral traction condition and the three-dimensional (3D) field equation in a pointwise manner. The closed system consists of 10 vector unknowns, and further refinements through elaborated calculations are performed to extract bending and torsion terms and to obtain recursive relations for the first- and second-order displacement coefficients. Eventually, a system of four asymptotically consistent rod equations for four unknowns (the three components of the central-line displacement and the twist angle) are obtained. Six physically meaningful boundary conditions at each edge are obtained from the edge term in the 3D virtual work principle, and a one-dimensional rod virtual work principle is also deduced from the weak forms of the rod equations.

On a consistent rod theory for a linearized anisotropic elastic material II. Verification and parametric study

“…The current paper is a subsequent work of [1] where a consistent rod theory was developed through an asymptotic reduction method for a rod composed of linearized anisotropic material. In the following, we briefly review this rod theory.…”

“…The key idea is the establishment of recursive relations which represent higher-order coefficients in terms of lower-order ones to eliminate most of the unknowns and this can be done just through linear algebraic equations. The basic framework and fundamental idea in [1] were developed by one of the present authors and his co-authors in [2–4] and several subsequent reduction theories have also been established by using this procedure, see [5–8]. Compared with these works [5–8], the feature of [1] is, as a three-dimensional (3D) to one-dimensional (1D) reduction theory, it creatively utilizes some special techniques (such as the Fourier series expansion of the lateral traction) to overcome some non-trivial challenges (such as the inconsistent and unclosed system of equations), which finally leads to a simple and physically meaningful rod theory including four scalar leading rod equations, recursive relations for higher-order coefficients, and six boundary conditions at each end through 3D virtual work principles.…”

“…Compared with these works [5–8], the feature of [1] is, as a three-dimensional (3D) to one-dimensional (1D) reduction theory, it creatively utilizes some special techniques (such as the Fourier series expansion of the lateral traction) to overcome some non-trivial challenges (such as the inconsistent and unclosed system of equations), which finally leads to a simple and physically meaningful rod theory including four scalar leading rod equations, recursive relations for higher-order coefficients, and six boundary conditions at each end through 3D virtual work principles. Other rod/beam theories are reviewed in detail in [1] and are not discussed further here. In the following, we concentrate on the application of these rod/beam theories.…”

“…However, this type of study is necessary because it can provide suggestions for achieving desirable mechanical responses (e.g., stress distribution and material strength) through appropriate choice of elastic moduli. Therefore, in the current paper, we intend to conduct a systematic study to examine the influences of elastic moduli on the deformation of a rod based on the rod theory developed in [1]. On the one hand, the analytical solutions are constructed for the deformation including the displacement, stress, axial twist angle, and principal stress for both isotropic material and four anisotropic materials.…”

“…The remaining content of the paper is arranged as follows. In Section 2, we list main results of the rod theory developed in [1], including the leading-order rod equations, recursive relations, as well as boundary conditions at two ends. In Section 3, analytical solutions for five benchmark Saint-Venant’s problems are constructed and are compared with the available 3D exact solutions for verification of the rod theory in [1].…”