[Paper Review] Brittle Crack Roughness in Three-Dimensional Beam Lattices

B. Skjetne, Torbjørn Helle+1 more|arXiv (Cornell University)|May 25, 2005

Numerical methods in engineering4 citations

TL;DR

This study investigates crack roughness in three-dimensional elastic beam lattices using numerical simulations to model brittle fracture. It finds a universal roughness exponent of ζ = 0.59 ± 0.01 under strong disorder, consistent with small-scale experimental observations and suggesting a distinct scaling regime for slow, quasi-static crack growth in disordered media with a characteristic length scale.

ABSTRACT

The roughness exponent is reported in numerical simulations with a three-dimensional elastic beam lattice. Two different types of disorder have been used to generate the breaking thresholds, i.e., distributions with a tail towards either strong or weak beams. Beyond the weak disorder regime a universal exponent of 0.59(1) is obtained. This is within the range 0.4-0.6 reported experimentally for small scale quasi-static fracture, as would be expected for media with a characteristic length scale.

Motivation & Objective

• To investigate the scaling behavior of crack roughness in three-dimensional elastic beam lattices under varying disorder.
• To determine whether the roughness exponent ζ exhibits universality in 3D elastic beam networks, particularly under strong disorder.
• To compare simulation results with experimental data and assess the relevance of the beam model to real brittle fracture processes.
• To explore the existence of distinct scaling regimes—particularly low ζ values at small length scales—consistent with experimental observations.
• To evaluate the role of characteristic length scales in fracture morphology, especially in systems with discrete elastic elements like beam lattices.

Proposed method

• Simulations use a cubic lattice of linearly elastic beams with fixed connectivity, where each beam resists axial, shear, bending, and torsional forces.
• The mechanical equilibrium is solved iteratively using the conjugate gradient method to minimize elastic energy, ensuring force and moment balance at each node.
• Breaking thresholds are assigned via a power-law disorder model: r^D, where r ∈ [0,1], with D controlling the strength and tail direction (weak or strong beams).
• Fracture is initiated by imposing a uniform displacement on the top surface, and beams break sequentially based on their weakest axial strength, with stress redistribution after each break.
• The roughness exponent ζ is computed from the root-mean-square fluctuation of the fracture surface height perpendicular to the average fracture plane, using W ∼ L^ζ.
• System size L ranges from 3 to 40, and the scaling exponent is extracted via linear fits to log-log plots of W vs. L for different D values.

Experimental results

Research questions

• RQ1 Does the roughness exponent ζ in three-dimensional elastic beam lattices converge to a universal value under strong disorder?
• RQ2 How does the roughness exponent ζ depend on the type of disorder, particularly when the threshold distribution favors weak or strong beams?
• RQ3 Is the observed roughness exponent consistent with experimental values reported for small-scale, quasi-static fracture?
• RQ4 Does the beam lattice model exhibit a distinct scaling regime at small length scales, as suggested by previous studies?
• RQ5 How does the beam model’s behavior compare to scalar models like the random fuse model or the Born model in predicting fracture roughness?

Key findings

• For strong disorder (D = 4), the roughness exponent converges to ζ = 0.59(1), indicating a universal scaling behavior in three-dimensional beam lattices.
• The exponent ζ = 0.59(1) is lower than the experimentally reported value of ζ = 0.8 for large-scale fracture but falls within the range 0.4 ≤ ζ ≤ 0.6 observed in small-scale, quasi-static experiments.
• For D = 2, the exponent is ζ = 0.62(2), and for D = 1, it is ζ = 0.77(1), indicating that the system only enters the universal regime under strong disorder.
• The result for D = -4 (ζ = 0.65(2)) is slightly higher but still consistent with the universal value, suggesting the transient regime for negative D may extend to higher |D| in 3D.
• The beam model’s ability to preserve rotational invariance and model realistic elastic deformation leads to a roughness exponent that aligns with small-scale experimental data, supporting the existence of a distinct scaling regime.
• The findings support the hypothesis that systems with a characteristic length scale—such as beam lattices—exhibit a lower roughness exponent at small scales, consistent with Cosserat elasticity and small-scale fracture behavior.