Asymptotically exact dimension reduction of functionally graded anisotropic rods
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908977355292672 |
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| author | Le, Khanh Chau |
| author_facet | Le, Khanh Chau |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically exact dimension reduction of functionally graded anisotropic rods Le, Khanh Chau Classical Physics 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. |
| title | Asymptotically exact dimension reduction of functionally graded anisotropic rods |
| topic | Classical Physics |
| url | https://arxiv.org/abs/2512.21483 |