Asymptotically exact dimension reduction of functionally graded anisotropic rods

TL;DR

The paper develops a rigorously reduced 1D theory for functionally graded, arbitrarily anisotropic rods by applying the variational-asymptotic method (VAM) to the 3D elasticity problem. It introduces dual cross-sectional problems to obtain provable upper and lower bounds on the effective 1D stiffnesses and uses the Prager–Synge identity to prove energetic asymptotic exactness, including dynamic validity through dispersion comparisons with 3D solutions. A systematic procedure restores full 3D stress and strain fields from the 1D solution, enabling detailed local analysis within an efficient framework. Numerical results for transversal isotropy and rhombohedral symmetry reveal meaningful transverse corrections and coupling effects that standard 1D theories miss. The framework supports the design of FG anisotropic components and waveguides, with potential extensions to multiphysics FG materials.

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. Furthermore, the dynamic validity of the theory is demonstrated 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.

Figures (8)

• Figure 1: A rod.
• Figure 2: Normalized torsional stiffness $\hat{g}$ as a function of the aspect ratio $a$: (i) at fixed $\delta=1$, $\gamma_U=1$ and four different $\gamma_L=0.2,0.4,0.6,0.8$ (left), and (ii) at fixed $\gamma_L=0.1$, $\gamma_U=0.4$ and four different $\delta=1,2,3,4$ (right).
• Figure 3: Numerical solution of the stress function $\check{\psi}$ for the anti-plane problem ($a=1$, $\delta=2$, $\gamma_U=1$, $\gamma_L=0.8$). The distribution characterizes the torsional shear stress state across the functionally graded cross-section.
• Figure 4: Percentage contribution of the transverse bending stiffness $\eta_{22}$ as a function of the right-side Poisson's ratio $\nu_R$: (left) for fixed $a=10$, $\delta=4$, $\nu_L=0.3$, and varying $\epsilon=0.2,0.4,0.6,0.8$; (right) for fixed $a=10$, $\epsilon =0.5$, $\nu_L=0.3$, and varying $\delta=1,2,3,4$. The remaining parameters are: $\epsilon_{1L}=2\epsilon$, $\epsilon_{1U}=2$, $\nu_{1L}=0.3$, $\nu_{1U}=0.4$.
• Figure 5: Percentage contribution of the transverse cross stiffness $\eta_{2}$ as a function of the right-side Poisson's ratio $\nu_R$: (left) for fixed $a=1$, $\delta=4$, $\nu_L=0.3$, and varying $\epsilon=0.2,0.4,0.6,0.8$; (right) for fixed $a=1$, $\epsilon =0.5$, $\nu_L=0.3$, and varying $\delta=1,2,3,4$.The remaining parameters are: $\epsilon_{1L}=2\epsilon$, $\epsilon_{1U}=2$, $\nu_{1L}=0.3$, $\nu_{1U}=0.4$.