Asymptotically exact dimension reduction of functionally graded anisotropic rods

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 $20\%$ in deflection predictions, the current VAM framework reduces this discrepancy to below $3\%$, with log-log convergence studies confirming the theoretical $O(h/L)$ accuracy.