The fracture resistance of elastic networks increases with the density of defects like a random walk

Technical Summary: The Fracture Resistance of Elastic Networks Increases with the Density of Defects Like a Random Walk

Problem Statement
Disordered spring networks serve as a fundamental model for studying fracture in diverse materials, ranging from ceramics and polymer networks to mechanical metamaterials. A central quantity in fracture mechanics is the apparent fracture energy, $G_c$, which measures the resistance to the propagation of a pre-existing dominant crack. While it is established that disorder can increase $G_c$ through crack arrest by local inhomogeneities, the quantitative dependence of this toughening on the degree of disorder (specifically defect density) remains poorly understood. Previous work has shown that in the absence of a pre-existing crack, defects generally promote nucleation and lower fracture resistance. However, in pre-cracked systems, the relationship is more complex: at low defect densities, increasing disorder may enhance resistance to crack propagation, whereas at high densities, the material weakens. The specific mechanisms linking microscopic defect statistics to macroscopic fracture energy in the regime of pre-existing cracks require further theoretical and quantitative clarification.

Methodology
The authors employ a two-dimensional triangular network of linear springs under quasi-static loading to investigate the effect of randomly placed missing bonds on crack propagation.

• Model System: The network consists of springs with rest length $\ell_r$ and stiffness $k$. A spring breaks abruptly when its strain exceeds a threshold $\varepsilon_{max} = 1$.
• Defect Introduction: A fraction $\nu$ of bonds is randomly removed from an otherwise perfect lattice.
• Simulation Protocol: The system is loaded via uniform vertical displacement increments. The network is relaxed to static equilibrium using the FIRE minimization algorithm. Springs exceeding the strain threshold are removed sequentially, starting from the most stretched, following a specific protocol to simulate crack propagation.
• Key Metrics: The apparent fracture energy $G_c(a)$ is defined as the critical elastic energy release rate required to propagate the crack from an initial length $L_c$ to $L_c + a$. The study focuses on the initiation energy $G_c(0)$ and the failure energy $G_c(a_f)$ at a specific crack advance $a_f$.
• Theoretical Framework: The authors map the elastic heterogeneity caused by missing bonds onto an equivalent local fracture energy landscape, $\Gamma_{loc}(a)$. They apply established theories of planar crack arrest, assuming that the apparent fracture energy at any point is determined by the maximum local fracture energy encountered along the crack path up to that point.

Key Results

1. Toughening via Defects: For a fixed concentration of missing bonds $\nu$, the apparent fracture energy $G_c(a)$ increases with crack advance $a$. Unlike perfect networks which fail instantaneously upon the first bond breaking, disordered networks exhibit intermittent crack propagation characterized by alternating unstable growth and arrest.
2. Scaling with Defect Density: The standard deviation of the fluctuations in the equivalent local fracture energy $\Gamma_{loc}$ increases with the square root of the missing bond fraction, $\sqrt{\nu}$. Consequently, the average failure fracture energy $\langle G_c(a_f) \rangle$ also scales with $\sqrt{\nu}$.
3. Random Walk Mechanism: This $\sqrt{\nu}$ scaling is explained by modeling the perturbations caused by individual missing bonds as independent random variables. The total perturbation to the local fracture energy is the superposition of these individual effects, analogous to a random walk. The variance of the total perturbation grows linearly with the number of missing bonds, leading to a standard deviation proportional to $\sqrt{\nu}$.
4. Exponential Tail and Logarithmic Growth: The probability density function (PDF) of the local fracture energy $\Gamma_{loc}$ exhibits an exponential tail rather than a purely Gaussian one. This exponential tail leads to a Gumbel extreme-value distribution for the apparent fracture energy. As a result, the median of $G_c(a)$ increases logarithmically with crack advance $a$ ($G_c \propto \ln a$).
5. Dominance of Crack Arrest: The study demonstrates that crack arrest by local fluctuations in $\Gamma_{loc}$ is the dominant toughening mechanism in the studied regime. While crack paths can deviate slightly from straight lines (roughness), the planar crack arrest model accurately predicts the observed behavior, suggesting that path roughness plays a minor role compared to the local heterogeneity-induced arrest.

Significance and Claims
The paper claims to establish a quantitative link between microstructural disorder and macroscopic fracture energy. By demonstrating that the fluctuations in local fracture energy scale as $\sqrt{\nu}$ due to the random-walk-like superposition of perturbations from individual defects, the authors provide a theoretical framework for predicting the toughening of disordered materials. The identification of the exponential tail in the local energy landscape explains the logarithmic increase of fracture energy with crack advance, a behavior distinct from systems without pre-existing cracks.

The authors note that these findings are expected to extend to other lattice structures and continuum systems where missing bonds correspond to microvoids or microcracks, provided the defect density remains low and fracture proceeds along a nearly straight path. They explicitly state limitations, including the assumption of planar crack propagation and the independence of bond perturbations, acknowledging that these assumptions may break down at very high defect densities or in three-dimensional systems where crack front averaging occurs. The work aims to pave the way for quantitative predictions of fracture energy in a wide variety of disordered materials by connecting microscopic defect statistics to macroscopic toughness.