[Paper Review] How Geometry Tames Disorder in Lattice Fracture

Matthaios Chouzouris, Leo de Waal+4 more|arXiv (Cornell University)|Feb 10, 2026

Microstructure and mechanical properties0 citations

TL;DR

The paper shows that lattice fracture in pre-cracked triangular beam lattices with Weibull-distributed failure thresholds exhibits three regimes of disorder expression, controlled by Weibull modulus and a Slenderness Ratio, and provides a mechanically informed statistical model predicting damage evolution and regime transitions.

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.

Motivation & Objective

• Understand how quenched disorder in beam-lattice fracture depends on geometry.
• Identify regimes of disorder expression: suppressed, localized near crack tip, and globally diffuse failure.
• Develop a mechanistically informed statistical framework linking disorder to damage metrics and crack-path morphology.

Proposed method

• Extend the Random Beam Model by discretizing each beam into three elements to capture bending-dominated deformation.
• Use Weibull-distributed failure stresses for beams to model disorder with modulus n.
• Derive a mechanically informed statistical framework that links SR (slenderness ratio) and n to probabilities of anomalous failures (scattering) and diffuse failure.
• Define crack-tip stress hierarchies as a function of SR to predict failure sequences and crack-path tortuosity.
• Derive P_s(n, λ) = κ_s^n / (1 + κ_s^n) for crack-tip scattering probability and P_d^(0) for initial diffuse failure from Weibull renormalization.
• Validate predictions with lattice fracture simulations across SR values and disorder levels.

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

Experimental results

Research questions

• RQ1 How does geometry (Slenderness Ratio) influence the expression of quenched disorder in lattice fracture?
• RQ2 What are the regimes of fracture behavior as a function of Weibull modulus and SR, and how can they be predicted theoretically?
• RQ3 How does disorder affect crack-tip damage versus diffuse damage, and how does this relate to crack-path morphology and toughness?
• RQ4 Can a statistically grounded framework predict the number of excess broken bonds and diffusion-like damage at fracture onset?
• RQ5 What role does the Weibull stress renormalization play in linking micro-scale failures to macro-scale fracture outcomes?

Key findings

• Three distinct failure regimes emerge: disorder effectively suppressed, disorder localizes near the crack tip altering crack morphology, and disorder manifests globally causing diffuse failure.
• Geometry via the Slenderness Ratio reshapes crack-tip stress hierarchies and controls the likelihood of anomalous (scattering) damage events.
• A mechanically informed Weibull-based framework predicts scattering probability P_s(n, λ) and diffuse-failure probability P_d^(0) that agree with simulations, including non-monotonic disorder-toughening behavior.
• Disorder-induced toughening is non-monotonic in disorder strength and dependent on lattice geometry, challenging interpretations based solely on crack tortuosity or diffuse damage.
• The SR can be used as a design parameter to control how disorder is expressed during fracture, enabling geometry-driven tuning of fracture behavior in lattice metamaterials.
• Simulation results for excess broken bonds and diffuse failure probabilities closely track theoretical predictions for weak to moderate disorder.

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 functio