Complexity of crack front geometry enhances toughness of brittle solids

The propagation and growth of cracks in brittle solids is a crucial area of research due to its implications for material failure and structural integrity. Understanding and predicting crack formation and fracture is essential for safety and cost-effectiveness in various engineering applications. Linear Elastic Fracture Mechanics (LEFM) is a widely used model for crack propagation, but it relies on the simplifying assumption that cracks are primarily planar. This assumption contradicts observations of real-world crack surfaces, which often exhibit complex three-dimensional (3D) geometries, including corrugations, textures, lances, ridges, microbranches, and crack nets. These deviations from planarity arise from factors like mixed-mode loading conditions and the inherent microstructure of the material. While LEFM can be adapted for non-planar cracks given known geometry, existing 3D perturbation methods are inadequate for dealing with the substantial deviations from planarity observed in real-world cracks. Moreover, there's a lack of universally accepted criteria for path selection in 3D crack propagation, despite the tendency of cracks in polycrystalline solids to propagate between crystal grains. Previous research has shown that step-like features can enhance energy dissipation during crack propagation, but comprehensive 3D data on complex crack tips is lacking. This paper aims to address this gap by using advanced imaging techniques to experimentally characterize the relationship between crack front geometry and toughness in brittle solids.

The authors extensively review existing literature on crack propagation and fracture mechanics. They discuss the limitations of linear elastic fracture mechanics (LEFM) when applied to non-planar cracks. The literature review highlights previous work on various crack surface features such as lances, ridges, mist, hackle, microbranches, and crack nets. It also mentions research on instability under mixed-mode loading conditions and the challenges associated with path selection criteria in three-dimensional crack propagation. The authors cite studies demonstrating that step-like features can enhance energy dissipation during crack growth, but underscore the lack of in-situ three-dimensional data on complex crack fronts, which prompted the current study to experimentally examine this aspect of fracture mechanics.

The researchers employed optical imaging methods, specifically confocal microscopy, to obtain high-precision, in-situ 3D crack tip kinematic data. Several brittle materials were used, including polyacrylamide hydrogels with varying gel chemistries and a polydimethylsiloxane (PDMS) elastomer. Fluorescent dyes were incorporated into the materials for improved contrast during imaging. An edge crack was introduced into each sample, and the sample was subjected to remote mode-I tensile loading while being imaged by confocal microscopy. Small strain increments were applied, and image stacks were recorded at each step. This allowed the researchers to track the evolution of the crack tip geometry up to the point of crack propagation. The crack tip opening displacement (CTOD) was measured to characterize the stress field at the crack tip. The geodesic length of the crack front was directly measured from the 3D image data. The researchers normalized the geodesic crack front length (ℓ) by the sample thickness (w) to obtain a dimensionless quantity ξ. Error assessment for the measurement of ξ, considering confocal microscope resolution and image segmentation, is detailed in the supplementary methods. The apparent fracture energy (Γ_app) was calculated from the CTOD data using linear elastic fracture mechanics relations. The total fracture energy (Γ) was partitioned into two components: the process zone fracture energy (Γ_pz) and the crack tip fracture energy (Γ_tip). A phenomenological expression was developed to relate the normalized strain energy release rate to the geodesic crack length. The study also included an experiment where a rigid Nylon particle was embedded in the hydrogel to induce crack plane symmetry breaking. The method involved comparing the loading state (assessed from CTOD) and the crack front geometry (from geodesic length) to investigate the relationship between these quantities. Image processing involved multiple software packages (Imaris, Fiji, scikit-image) and a detailed workflow to segment the 3D image data.

The study revealed a direct, linear relationship between the critical strain energy release rate (G_c) and the normalized geodesic crack front length (ξ) across various materials. This linear relationship indicates that increasing the complexity of the crack front geometry increases the material's toughness. The constant of proportionality in the linear relationship was found to be less than expected based on pure energy dissipation along the crack front, suggesting the presence of a diffuse damage region accompanying the localized dissipation at the crack front. The analysis showed localized crack advance along the crack front before global crack propagation, indicating that the crack can be locally critically loaded and advance fractionally out of the crack plane without causing global failure. The researchers found that the total fracture energy (Γ) could be partitioned into process zone energy (Γ_pz) and crack tip energy (Γ_tip), with Γ_tip scaling linearly with the normalized crack front length. The experiment involving a rigid inclusion demonstrated that even induced crack plane symmetry breaking, resulting from interactions with inclusions, leads to a significant increase in G_c. This observation can be integrated into the existing G_c vs. ξ relationship. The study provides quantitative evidence showing that complex crack front geometry systematically enhances the material toughness. This enhancement is due to a non-uniform critical loading state along the crack front, enabling local instability while maintaining global stability until catastrophic failure.

The findings of this study directly challenge the assumptions of linear elastic fracture mechanics (LEFM), which typically models cracks as planar. The observed linear relationship between critical strain energy and geodesic crack length suggests that accounting for crack front complexity is essential for accurate prediction of material toughness. The partitioning of fracture energy into process zone and crack tip contributions provides a mechanistic explanation for the observed toughness enhancement. The observation of fractional crack front advancement, where local instability coexists with global stability, introduces a new perspective on the physical conditions governing 3D crack growth. The results have important implications for materials testing, as deviations from a planar crack front can lead to mismeasurement of material toughness. The study also opens avenues for manipulating crack front geometry to enhance toughness, which could be relevant in the design of composite materials and other engineering applications.

This research demonstrates that the complexity of crack front geometry significantly enhances the toughness of brittle solids. The critical strain energy required for crack propagation is directly proportional to the geodesic crack length, suggesting that geometric complexity can be exploited to improve material strength. This finding challenges the assumptions of LEFM and highlights the need for more sophisticated theoretical models of 3D crack propagation. Future research should focus on developing theoretical frameworks that incorporate crack front geometry, investigating the limits of geometric toughening, and exploring strategies for manipulating crack front complexity to design tougher materials.

The study primarily used transparent hydrogel and elastomer materials, which may not fully capture the behavior of opaque or more complex materials. The phenomenological expression relating strain energy release rate to geodesic crack length is based on a specific energy partitioning model, and further investigation might be needed to validate this model for various materials. The study did not exhaustively evaluate all candidate geometric expressions for the relationship between fracture energy and crack front complexity.