How Geometry Tames Disorder in Lattice Fracture

TL;DR

The paper addresses how quenched disorder affects fracture in beam-lattice metamaterials and demonstrates that lattice geometry—captured by the Slenderness Ratio—controls how disorder manifests. A mechanically informed statistical framework, built on Weibull statistics and crack-tip micromechanics, predicts three distinct failure regimes and a non-monotonic disorder-induced toughening, which is validated by extensive simulations. The results reveal that apparent toughness gains arise from statistical local fluctuations and crack-arrest phenomena rather than simply increased crack tortuosity, offering design principles for geometry-driven control of fracture in architected materials. Overall, the study shows how geometry can actively regulate disorder expression, enabling disorder-tolerant and tunable fracture responses in engineered lattices.

Abstract

We investigate the fracture behavior of pre-cracked triangular beam-lattices whose elements have failure stresses drawn from a Weibull distribution. Through a statistical analysis and numerical simulations, we identify and verify the existence of three distinct failure regimes: (i) disorder is effectively suppressed, (ii) disorder manifests locally near the crack tip, modifying the crack morphology, and (iii) disorder manifests globally, leading to initially diffuse failure. Our model naturally reveals the key parameters governing this behavior: the Weibull modulus, quantifying the spread in failure thresholds, and a geometric quantity termed the Slenderness Ratio. We also reproduce the disorder-induced toughening reported in previous experimental and numerical studies, further demonstrating that its manifestation depends non-monotonically on disorder. Crucially, our results indicate that this toughening cannot be simply connected to the amount of damage in the lattice, challenging interpretations that attribute increased fracture energy solely to enhanced crack tortuosity or diffuse failure. Overall, our results establish geometry as a powerful control parameter for regulating how disorder is expressed during fracture in beam-lattices, with broader implications for the disorder-induced toughening in engineered materials.

Figures (10)

• Figure 1: Problem setup. (a): We consider rectangular domains of triangular lattices, with a half domain horizontal crack along the vertical midpoint, pinned to the horizontal, and with roller constraints along the vertical boundaries. The horizontal boundaries are kept parallel and moved apart to stretch the lattice and induce damage. The most stressed bond under these loading conditions is shown by the red arrow and colored blue. (b-d): Visualization of the lattice stresses just before and long after (insets on the top right of each) the onset of failure, in quasi-static loading. Elements in the lattice are colored according to their proximity to failure, which is the ratio of their combined axial-bending stress ($\sigma$) to their individual failure stresses ($\sigma_f$); insets on the left show the respective Weibull survival distributions. Increasing disorder (smaller $n$) progressively hides information about the elements' stresses.
• Figure 2: Micromechanics and damage evolution after an anomalous failure event. (a): Crack-tip stress hierarchy and its geometric control. Top: Ratios of the maximum stresses in each of the six crack-tip beams to that of the most highly stressed beam (element 1, shown in blue), plotted as a function of the Slenderness Ratio ($\lambda$). Solid lines correspond to the baseline failure criterion, with ultimate axial and bending stresses taken as $45$ MPa and $85$ MPa, respectively. Faint lines for elements 5 and 6 show the modified stress ratios obtained when the ultimate bending and axial stresses are taken to be equal (at $45$ MPa), illustrating how changes in bending failure strength alter the relative criticality of crack-tip elements. Bottom: Relative contributions of axial and bending stresses to the total stress in each crack-tip beam as a function of SR. (b): Resulting crack-path after removing each one of the crack-tip elements, at $\lambda = 10$, and no disorder ($n \rightarrow \infty$). The red dots depict split nodes, while the circled numbers show the order of subsequent failure events. Observe that subplots iii-vi, contain one more broken element within their lower frame compared to i-ii, owing to the additional bond the crack had to remove in order to transverse the same horizontal distance. Beam thickness is not drawn to scale.
• Figure 3: Renormalization of a set $\mathcal{G}$ of $m$ elements. The resulting element can be regarded to be at a Weibull stress $\bar{\sigma}_{\mathrm{ \mathcal{G}}} \geq max(\{\sigma_1, \sigma_2, ... \sigma_m \})$, i.e., larger than any of the stresses of its constituents. The resulting element survives iff all of its constituents also survive.
• Figure 4: Fracture regimes resulting from the mechanically informed statistical analysis. At each point in parameter space, the most probable mode of non-horizontal crack propagation is used to color the corresponding square according to eqs. (\ref{['eq:non_adj_frac']}) & (\ref{['eq:p_scatter']}), with the opacity indicating the associated probability. This construction naturally gives rise to three distinct regions, which are labeled accordingly. Insets show representative failure patterns drawn from each region. Dashed lines are shown as guides to the eye, and delineate the different regimes identified; the conditions defining these boundaries are indicated in text over the dashed lines. At fixed disorder strength, varying the SR alone is sufficient to drive transitions between these regimes, highlighting the possibility for controlling crack-path morphology through geometry.
• Figure 5: Simulation data plotted over theoretical predictions for the probabilities of scattering and diffuse failure. Note that the theoretical (dashed) lines are slices of the respective probabilities from Fig. \ref{['fig:Figure4']}, and they have no free parameters; i.e., they are overlaid, not fitted to the data. Error bars indicate the standard error of the mean. (a): Ratio of excess broken bonds $\Delta N = N_f - N_x$ to $N_x$, over the theoretical predictions for $P_s$. In the inset we show the same data in semi-log scale, which clearly demonstrates not only convergence, but also the exponential form for the decay of probability $P_s$ at large $n$ (b): Probability of diffuse failure at the beginning of the damage process---${P^{(0)}_d}$---, with theoretical prediction overlaid. The weak dependence on SR of this probability is obvious both for theory and simulation data. The inset shows the mean fraction of diffuse failure points as a function of the mean size of the main cluster $N_f$, for $\lambda = 10$, with the dashed lines being fits of Eq. (\ref{['eq:diffuse_failure_evolution']}); ${P^{(0)}_d}$ in the main plot is extracted as the y-intercept of these curves.