Diffusive buckling fronts in lattice-based metamaterials

Abstract

Mechanical metamaterials can be designed to exhibit unique mechanical properties, including tunable auxetic behavior as well as multi-stability, which arise from the geometry and configuration of the constituent building blocks. Lattice-based metamaterials, in particular, provide lightweight platforms where local instabilities can dictate the global response, with applications in energy routing, vibration isolation, and impact mitigation. In underdamped structures, perturbations have been found to propagate as nonlinear waves, e.g., transition waves or solitons. Here we investigate the opposite limit of overdamped, highly dissipative lattice metamaterials. Focusing on three-dimensional structures, we uncover how buckling instabilities, triggered by compression, propagate as fronts that shape the macroscopic behavior. We demonstrate in experiments on 3D-printed simple cubic lattices how global and local buckling modes can be controlled via the lattice geometry. By incorporating viscoelastic dissipation into a 3D-continuum model, we show that strain-driven buckling fronts obey coupled reaction-diffusion equations. The diffusion and reaction coefficients, determined by local geometry, material properties, and strain, select the propagation direction and enable steering of the fronts. This establishes a predictive and experimentally validated framework for the control of cascading mechanical instabilities in lattice-based metamaterials.

Figures (8)

• Figure 1: 3D-printed cubic lattices under compression. (A) 3D-printed cubic lattice with rigid nodes (white) and flexible bonds (lightblue) where $N = 5$, $L = 30\mm$, $R_{\mathrm{node}} = 2.5\mm$, $R_{\mathrm{bond}} = 0.75\mm$ and $L_{\mathrm{bond}} = 2.5\mm$. (B) Sequence of images (front view) showing the dynamical response of two slightly different lattice structures to uniaxial compression, see Suppl. Movie 1. The overall response transitions from local to global buckling once a critical strain, $\epsilon_{\mathrm{crit}}$, is reached. The rightmost panel shows the difference in self-folding of the five layers from the side. The white dotted lines indicate where the bottom of the compression plate touches the top of the lattice. (C) Node rotation $\theta$ during local and global buckling of the nine center nodes, showing counter-rotating neighbours (upper) and co-rotating rows (lower), respectively. Clockwise rotations are defined positive (see (A)). (D) Front- and side view of a compressed cubic lattice with $N = 4$, $R_{\mathrm{node}} = 2.63\mm$, $R_{\mathrm{bond}} = 0.75\mm$, and $L = 22.5\mm$ at $\epsilon = 0.125$, see Suppl. Movie 2. Within different layers of the cube the orientation of the pattern formation changes (green regions). This leads to torsion of the bonds (red region) connecting these layers and the formation of a domain wall. (E) Parameter space in terms of bond length and bond radius indicating the cubic lattices that buckle globally (blue triangles) or locally (green diamonds), including those that show the formation of a domain wall during local buckling (red squares).
• Figure 2: Buckling phase space and mechanical response. (A) Parameter space indicating the regions of global (blue triangles) and local (green diamonds) buckling. The gray area is beyond the available design space. The red line corresponds to Eq. \ref{['Eq:criterion']} and its slope is the ratio of the effective Young's and shear moduli of the lattice material $E_{\mathrm{bond}}/G_{\mathrm{bond}} = 4$. (B) Poisson's ratio averaged over the center nine nodes $\bar{\nu}$ as a function of applied strain $\epsilon$ for the 'front' view. Dark-blue to red, $R_{\mathrm{bond}} = [2\mm, 1.25\mm, 1\mm, 0.75\mm]$. (C) Reaction force $F$ during compression as a function of strain $\epsilon$ for four different lattice designs with $R_{\mathrm{node}} = 2.5\mm$. The first local maximum indicates $F_{\mathrm{crit}}$ and $\epsilon_{\mathrm{crit}}$ (red triangles). The initial slope (dashed line) gives the effective Young's modulus $E_{\mathrm{eff}} \sim \mathrm{d}F/\mathrm{d}\epsilon$ of the lattice. (D) Normalized critical buckling force of 15 different designs as a function of material properties on a double-logarithmic scale. Each point represents one experiment and its color reflects the node radius. The inset shows the measured relation between $K$ as a function of $R_{\mathrm{node}}$ for these lattices.
• Figure 3: Definitions for 3D-continuum model. (A) 3D-schematic indicating different xz-planes (red and blue regions) and the corresponding $\theta$ and $\phi$ node rotations in the xz- and yz-plane, respectively. The box indicates bonds of the cubic lattice that are modelled as axial (compression and shear) or torsional springs (bending and torsion). All bonds experience viscoelastic dissipation. (B) Schematic of a sliced portion of the cubic lattice showing the definitions for $\theta$ and $\phi$ in their corresponding planes, as well as the indexing of the nodes. We assume a counter-rotating coordinate transformation within the plane, resembling local buckling.
• Figure 4: Buckling plane selection and propagation in 3D. (A) Cross-sections of the 3D domain showing the temporal evolution of the normalized node rotations $\tilde{\theta}$ and $\tilde{\phi}$ in the counter-rotating basis, given random initial angular perturbations. Eventually, a buckling plane is selected with $\tilde{\theta} \neq 0$ and $\tilde{\phi} = 0$, or vice versa. (B) Local buckling propagation through a 3D lattice structure given nucleation sides at the less constrained edges and random initial angular perturbations.
• Figure 5: Diffusive buckling fronts in 2D lattices. (A) Sketch of setup and sequence of images during compression, see Suppl. Movie 3 & 4. (B) Top: Overlayed node lab frame rotation during compression of the regular lattice. Bottom: 2D simulations with similar initial conditions showing the propagating buckling front in the counter-rotating basis. (C) Normalized averages of the rotation angles as a function of rescaled vertical coordinate $z^{\star}$ at different instances in time, for crosshead speed $0.5\mm \per \s$. (D) Same as (C) but for three different crosshead speeds, collapsed by normalizing time by strain rate. Different traces are for different strain. (E) Same as (B) but with thicker bonds in the red regions.