Scale-Rich Network-Based Metamaterials

TL;DR

Scale-Rich metamaterials introduce a minimal, two-parameter design framework that embeds multiscale heterogeneity into mechanical architectures by letting ligament thickness decay as $\lambda_t=\lambda_0 t^{-\alpha}$ and ligaments be added randomly until jammed or tunable-density states emerge. The SR model produces phase behavior with jammed and tunable-density regimes and generates power-law distributions in thickness, length, and degree (e.g., $P(k)\sim k^{-3}$ in 2D), yielding true multiscale connectivity and geometry. Through simulations and experiments, SR metamaterials exhibit highly tunable elastic anisotropy across densities, delocalized nonlinear deformation with enhanced energy absorption, and programmable acoustic wave control including Luneburg-like lenses and GRIN-inspired devices. The approach extends naturally to 3D via plates and yields universal degree-scaling laws, offering a universal, inverse-design-ready paradigm for multifunctional materials whose properties emerge from scale diversity.

Abstract

Materials, at their essence, are networks defined by homogeneity: uniform bonds, fixed thicknesses, and discrete length scales. Mechanical metamaterials, while representing structurally more diverse microstructures, remain defined by the homogeneity of their unit cells, pore sizes, or repeating features. In contrast, as network science has revealed, real-world and biological systems -- from the Internet to the brain -- derive their function from broad, multiscale variability in connectivity and link length. Here, we introduce Scale-Rich (SR) metamaterials, a design framework that embeds network heterogeneity into mechanical metamaterials, achieving order-of-magnitude heterogeneity in ligament lengths, thicknesses, and connectivity. Governed by only two parameters, SR networks span orders of magnitude in structural features, overcoming prior constraints in metamaterial design. Translating these network models into physically realizable materials, we use simulations and experiments to show that SR metamaterials exhibit properties inaccessible to traditional single-scale systems, including highly tunable elastic anisotropy, delocalized nonlinear deformation with high energy absorption, and programmable acoustic wave control. This network-science-based paradigm establishes a minimal yet universal framework for engineering multifunctional materials whose mechanical and acoustic behavior emerge directly from scale diversity itself.

Figures (43)

• Figure 1: Scale-Rich networks.(a) Generation process of an SR metamaterial, starting at time step $t=1$ where an initial ligament of thickness $\lambda_0$ is generated from a randomized coordinate $(x_0,y_0)$ in a randomized orientation $\theta_0$. Additional ligaments are then generated at each time step $t$ (nucleated with randomized coordinates and orientations) with thickness $\lambda_t = \lambda_0 t^{-\alpha}$, terminating at the intersection with another ligament or the domain boundary. (b) The network behind the SR metamaterial, inspired by road networks rosvall2005networks, represents ligaments as nodes and intersections as links. The panels show the growth of the network corresponding to the configurations shown in (a). (c) The analytically derived ($\alpha, \lambda_0$) phase diagram (two-parameter design space) of SR metamaterials. For small $\alpha$ the system gets jammed, i.e. it does not allow the introduction of new ligaments. For large $\alpha$, the system can indefinitely accommodate additional ligaments, reaching a phase where the density of the system is tunable (see SI \ref{['subsec: phase_diagram']}). Experimentally realized SR metamaterials with $T=500$ and relative density $\bar{\rho}=40\%$, via two types of 3D-printing techniques, are depicted with white symbols. (d) Images of 3D-printed SR metamaterial specimens, at centimetre (vat photopolymerization) and micrometre (two-photon lithography) scales, corresponding to the white symbols in (b). Distributions of (e), ligament thickness $P(\lambda)$; (f), ligament length $P(\ell)$; and (g), ligament degrees (number of connections per ligament) $P(k)$, for $\lambda_0 = 0.1$, all exhibit a power-law tail. (h) Relative density modulation as a function of $\alpha$ for $\lambda_0 = 1$, averaged over 20 runs for $T = 30000$. (k)--(m) Thickness, length, and degree distribution of SR metamaterials ($T=500$, $\alpha=0.5$, $\lambda_0=0.06$) compared to reference systems: hexagonal lattice, square lattice, and Voronoi tessellation (all at $\bar{\rho}=40\%$).
• Figure 2: Linear-elastic behavior of the Scale-Rich microstructure.(a) Directional stiffness plots of the reference hexagonal (i), Voronoi (ii), and square (iii) lattices from finite element simulation, compared with SR samples with $\alpha = 0, 0.3, 0.9$ at the same relative density (v)-(vii). Panels (vii)-(ix) show the experimentally measured directional stiffness for SR materials with $\alpha = 0, 0.1, 0.3$, validating the theoretically predicted stiffness anisotropy. Scale bars in (vii)-(ix) measure 10 mm. (b) Anisotropy-density plot, illustrating the broad range of anisotropy achievable with the SR framework compared to the narrow range of values attained by the reference samples. The black filled circles (vii)-(ix) show the experimentally measured anisotropy. (c) As the decay parameter $\alpha$ increases, anisotropy arises because the maximum specific directional stiffness increases, whereas the minimum specific directional stiffness is constant. The solid lines represent the mean across all samples, and the shaded regions correspond to a $\pm 1$ standard deviation from the mean value. (d) Phase diagram in $(\alpha, \lambda_0)$ space of the anisotropy, which functions as a design map for a tunable anisotropic response.
• Figure 3: Nonlinear behavior of the Scale-Rich microstructure.(a) Experimental stress–strain curves of Scale-Rich and reference samples, for $\bar{\rho}_{\mathrm{nom}} = 42\%$, under uniaxial compression. For the reference samples, the mean behavior of three replicate specimens is plotted. For the SR geometries, the solid line represents the mean of all samples with $\alpha = 0$, whereas the dashed curves are individual samples. The response of SR samples with $\alpha > 0$ is given in SI Fig. \ref{['fig:SI-high-alpha-stress-strain']}. (b) Experimental toughness-density Ashby plot, showing the energy absorption capacity across all relative densities. (c) Evolution of localized deformation with increasing macroscopic strain: still images together with computed areal Jacobians, $J = A/A_0$, for each cell. The geometries dominated by a characteristic length scale all develop regions of compaction where localization is observed; only the Scale-Rich sample avoids the formation of compaction bands. We tracked images up to a macroscopic strain of 25%, after which compaction precludes the accurate computation of Jacobian values. All scale bars measure 10 mm. (d) Evolution of the localization parameter $\mathcal{L}$ in each geometry together with snapshots of the deformation evolution as recovered from still images. Here, the mean and standard deviation of three unique SR specimens is plotted.
• Figure 4: Elastic lenses from SR metamaterials.(a) SR samples with their corresponding directional velocity polar plots, demonstrating a wide range of achievable space of wave velocities and anisotropy. Curves, matching the color of their respective geometries, denote the quasi-longitudinal wave velocities, while the green curve represents the wave velocity of the monolithic material. Samples in the gray-black color scale show the breadth of wave anisotropy achieved, while the samples in the rainbow scale demonstrate isotropic SR unit cells used to construct the elastic lens in (b). We use each sample as a unit cell, adding a frame with small gaps to provide boundary conditions for building the superstructure. The effective refractive index, $n_\mathrm{eff}$, is defined as the ratio of the green to blue curves along the $x$-direction. (b) One quarter of a Luneburg-inspired lens constructed from SR samples with varying refractive indices, highlighted by colour (blue, high refractive index; orange, low refractive index). (c) Schematic illustration of the steady-state wavefield of the SR elastic Luneburg-like lens showing the elastic perturbation plane source (white) and focal point (red). The elastic wave is focused on the side opposite the source, as evidenced by the radial wave pattern emanating from the focal point. On the left, the pixelated refractive index map and analytical solution from optical equations are shown below and above, respectively. (d) The normalized displacement intensity along the y-axis cross-section through the lens focal point (white dashed line in panel d). The Luneburg-like lens (solid blue curve) exhibits a sharp focal peak, whereas the randomized lens (inset, green dashed curve) shows no noticeable focusing. The residual intensity peak in the randomized lens—three orders of magnitude smaller—arises from weak energy trapping at the elastic circular boundary. A suitable excitation frequency was found in the range $f \sim 646.7$Hz, well within the long-wavelength regime of each pixel in the lens, where the refractive indices are still valid.
• Figure 5: Design versatility of the Scale-Rich framework.(a) The SR framework enables precise control over ligament orientation, allowing the generation of (i) random configurations where angles are selected from a uniform distribution, (ii) aligned networks using random selections from a prescribed set of angles, and (iii) polarized patterns by sampling angles based on a predefined vector field. (iv) Nucleation point positions can also be controlled to vary density across the domain, enabling gradient patterns. The model further generalizes to three dimensions through the growth of (v) plates and (vi) beams, where two angles specify the orientation of each plate or beam (for more examples, see SI \ref{['subsec: model_limit']}). Scale bars, 10 mm. (b) SR microstructure generated within a non-convex domain by combining three distinct orientation patterns---random, aligned, and polarized---each applied on a different region of the domain. (c) Functional design using the SR framework: a microstructure is chosen to fill a given domain under a relative-density constraint in an effort to protect a "payload". (d) The SR framework allows simultaneous variation of both the ligament density and orientation. A circular design inspired by the natural microstructure of Citrus maximale2023influence is shown, with a linear density gradient in the radial direction combined with a polarized orientation applied to 20% of the initial lines. (e) Stress reduction in SR-protected payloads. (i-iii) Disc-shaped payloads made of poly(methyl methacrylate) (PMMA) were characterized using photoelasticity to visualize stresses under nonlinear deformation. Square and hexagonal lattices generate multiple fringes, indicating high internal stress, whereas the SR architecture shows markedly fewer fringes. Scale bars, 10 mm. (v - vii) Nonlinear simulations confirm these findings: the SR design reduces the average stress on the embedded payload by a factor of up to $4.91$ compared to the reference geometries (iv).