Asymptotically Exact Dimension Reduction of Functionally Graded Anisotropic Rods

ABSTRACT This study utilizes the variational‐asymptotic method to establish a one‐dimensional theory for functionally graded rods characterized by general anisotropy from the three‐dimensional elasticity theory. A distinctive feature of this dimension reduction procedure is the numerical solution of dual cross‐sectional problems, which provide rigorous upper and lower bounds for the average transverse energy density. By employing the Prager–Synge identity, we derive an error estimate in the energetic norm to establish the asymptotic exactness of the model. This estimate is extended to the dynamic regime for low‐frequency vibrations. Furthermore, the dynamic validity of the theory is confirmed by comparing the one‐dimensional dispersion relations with exact analytical three‐dimensional solutions for wave propagation in composite rods. The results show that the developed one‐dimensional model captures the long‐wave asymptotic behavior of the three‐dimensional elastic body with high fidelity. Numerical benchmarks indicate that while the naive rod theory incurs errors up to in deflection predictions, the current VAM framework reduces this discrepancy to about , with log–log convergence studies confirming the theoretical accuracy.