Homogenization of architected materials incorporating shearable beams

TL;DR

This work extends 2D architected-grid homogenization to include shear via Timoshenko beam theory, enabling accurate modeling of stubby beams and multi-plane junctions. By deriving a unit-cell energy framework and matching it to a Cauchy continuum, the authors obtain an explicit, rigorous effective elasticity tensor that captures axial, shear, and bending interactions. The results show that allowing shear deformability expands the achievable elastic properties, including Poisson's ratio and isotropy, beyond what slender Euler–Bernoulli beams can realize, and reveal robust auxetic behavior in re-entrant and hexagonal lattices. Finite-element validation confirms the approach, and the findings offer practical design guidance for tailoring stiffness, anisotropy, and auxeticity in microstructured lattices using stubby or microstructured beams and cross-plane junctions.

Abstract

Two-dimensional architected materials are often realized as periodic grids of elastic beams. Conventional homogenization methods represent these structures as equivalent elastic solids but neglect shear deformation in the constituent beams. This article addresses this limitation by incorporating shear deformability through Timoshenko beam theory, enabling accurate modeling of stubby beams. Moreover, shearable beams with extreme mechanical characteristics can be obtained through the design of appropriate microstructures. Introducing shearable beams into the grid expands the design space, allowing, for instance, the control of the effective Poisson's ratio beyond the limits achievable with slender beams.

Figures (29)

• Figure 2: Effective elastic modulus $E$ and Poisson's ratio $\nu$ characterizing an elastic solid of out-of-plane thickness $h$, equivalent to a two-dimensional grid of beams arranged in a hexagonal lattice (inset). Two models are considered for the constituent beams: (i) a homogenized discrete chain, characterized by the slenderness $\Lambda=l \sqrt{k_a/k_r}$, where $k_a$, $k_t$ and $k_r$ are the stiffnesses of the axial spring, slider spring, and elastic hinge, respectively, and $l$ is the total chain length, see Fig. \ref{['trave_a_taglio']}; (ii) a continuous elastic beam of length $l$, cross-section area $A = b h$, slenderness $\Lambda=2\sqrt{3}\,l/b$, and shear correction factor $\kappa$, made of an isotropic elastic material with Young's modulus $E_s$ and Poisson's ratio $\nu_s$ (the grid is obtained by superimposing beams on different planes, as explained in Section \ref{['sec:decoupled_plane_stress']}). Different curves represent various ratios $k_a/k_t$ ($E_s A/(\kappa\, G_s A)$ for the continuous model), with dashed lines indicating the limiting case $k_t \to \infty$ (or $\kappa\, G_s A \to \infty$), which corresponds to the Euler–Bernoulli theory.
• Figure 3: Concept model of a nodal connection where four stubby beams are jointed. The connection is designed for two orthogonal rods as a multi-layer rigid joint (colored in purple), where the out-of-plane symmetry is preserved: (left) exploded view; (right) perspective view.
• Figure 4: Deformed configuration of the planar Timoshenko beam at coordinate $x$. The material point $\bm x(x) = x\,\bm e_1$ is mapped to its deformed position $\bm r(x)$. Vectors $\bm a$ and $\bm b$ are, respectively, orthogonal and tangent to the cross-section at $\bm r(x)$, as defined in Eq. \ref{['eq:normal_parallel_cross_section']}. The figure also illustrates the slope of the beam axis, $v'(x)$, the rotation of the cross-section, $\theta(x)$, and the shear deformation, $\gamma(x) = v'(x)-\theta(x)$.
• Figure 5: Undeformed (left) and deformed (right) configurations for a square periodic lattice and its unit cell, made up of Timoshenko beams, represented thick to emphasize the effects of shear deformation. In the right panel, the gray beams represent the undeformed configuration for reference.
• Figure 6: Effective shear modulus $\tilde{G}=\tilde{\mathbb E}_{1212}$ for a square grid of Timoshenko beams with rectangular cross-section, plotted as a function of the Poisson's ratio $\nu_s$ of the beam material (left) and as a function of the beam slenderness $\Lambda$ (right). In the limit of stubby beams ($\Lambda \to 0$), the Timoshenko model predicts a finite shear stiffness for $\nu_s \neq -1$, whereas the Euler-Bernoulli model (red dashed line) leads to an infinite value. The shear stiffness increases as $\nu_s$ becomes more negative, reaching a maximum in the limit $\nu_s \to -1$.