A uniform framework for the dynamic behavior of a rod composed of a linearized anisotropic material is constructed based on an asymptotic reduction method. Taylor–Young series expansions of the displacement vector and stress tensor are adopted at the beginning. By substituting the series expansions into the three-dimensional (3D) equilibrium equations and the lateral traction condition followed by proper manipulations, four scalar rod equations are obtained for the leading-order displacement vector and the twist angle of the central line. The associated one-dimensional (1D) boundary conditions are obtained through the virtual work principle. One key feature of the present theory is the establishment of recursive relations so that most of the unknowns from the expansion can be eliminated. Also, all the remainders are retained, so the pointwise error estimate of the equations is established. One important advantage of the present theory is that no ad hoc assumption on the forms of displacement and external loading is adopted. Therefore, the rod theory provides a uniform framework to study dynamic behaviors. As an illustrative example, the natural vibration of a linearized isotropic rod with fixed-free boundary conditions is studied. The natural frequency is calculated by applying this uniform framework for the bending, torsional, and longitudinal modes. The obtained results are compared with those of the 3D simulations, Euler–Bernoulli, and Timoshenko beam theories. It turns out that the present theory is accurate enough for moderate and long rods in the bending vibration, and provides explicit results with high accuracy for either long or short rods in torsional and longitudinal vibrations.
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.
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