Asymptotically exact theory of functionally graded elastic beams

TL;DR

The paper develops an asymptotically exact one-dimensional theory for functionally graded beams by applying the variational-asymptotic method to 3-D elasticity with cross-sectional property variation. It reduces the problem through decoupled cross-sectional analyses (plane-strain and anti-plane) solved via finite elements and dual formulations to produce an explicit 1-D energy density $\Phi$ and associated stiffnesses, including torsion through $C_\angle$. An error estimate based on the Prager-Synge identity shows the 3-D stress error is $O\left( h/L \right)$ in the energetic norm, establishing the method's reliability for slender beams. The results reveal substantial contributions from the average transverse energy for certain FG gradations and Poisson's ratio contrasts, enabling accurate 3-D stress recovery from the 1-D solution and enabling extension to curved beams and smart materials with accompanying Matlab code for cross-sectional analyses.

Abstract

We construct a one-dimensional first-order theory for functionally graded elastic beams using the variational-asymptotic method. This approach ensures an asymptotically exact one-dimensional equations, allowing for the precise determination of effective stiffnesses in extension, bending, and torsion via numerical solutions of the dual variational problems on the cross-section. Our theory distinguishes itself by offering a rigorous error estimation based on the Prager-Synge identity, which highlights the limits of accuracy and applicability of the derived one-dimensional model for beams with continuously varying elastic moduli across the cross section.

Figures (6)

• Figure 1: A beam
• Figure 2: Dependence of normalized torsional stiffness $\bar{c}$ on the normalized width $a$: (i) at fixed $\delta=1$, $\nu_L=0.1$, $\nu_R=0.4$ and four different $\kappa=0.2,0.4,0.6,0.8$ (left), and (ii) at fixed $\kappa =0.5$, $\nu_L=0.1$, $\nu_R=0.4$ and four different $\delta=1,2,3,4$ (right).
• Figure 3: Normalized torsional stiffness $\bar{c}$ as a function of the normalized width $a$ for fixed $\delta=0.5$, $\kappa=4$, $\nu_L=0.3$, and varying $\nu_R = 0.1, 0.2, 0.3, 0.4$.
• Figure 4: Normalized bending stiffness $\bar{e}_{22}$ as a function of the normalized width $a$: (a) for fixed $\delta=4$, $\nu_L=0.1$, $\nu_R=0.4$, and varying $\kappa=0.2,0.4,0.6,0.8$; (b) for fixed $\kappa =0.5$, $\nu_L=0.1$, $\nu_R=0.4$, and varying $\delta=1,2,3,4$.
• Figure 5: Normalized bending stiffness $\bar{e}_{22}$ as a function of the normalized width $a$ for fixed $\delta=4$, $\kappa=0.5$, $\nu_L=0.3$, and varying $\nu_R = -0.9, -0.5, -0.1, 0.3$.