Elastic self-averaging does not ensure representative failure behavior in field-constrained basalt fracture networks

Deep underground, where massive caverns are carved for hydropower stations, the rock is rarely a solid, unbroken block. Instead, it is a complex puzzle of stone fragments held together by a network of cracks, faults, and joints. Engineers need to know how this fractured rock will behave when they dig into it or place heavy loads upon it. To make these predictions, they rely on a concept called the "representative elementary volume." In simple terms, this is the minimum size of a rock sample needed to accurately represent the behavior of the entire rock mass. If a sample is too small, it might miss a critical crack and give a false sense of security; if it is too large, the calculations become impossibly expensive. For decades, the standard assumption has been that once a rock sample is large enough to show a stable average stiffness, it is large enough to predict everything else, including how and when the rock will fail.

A new study challenges this long-held belief. Researchers from Sichuan University of Science & Engineering investigated whether a rock sample that looks stable in terms of its stiffness is actually stable in terms of its strength and ability to absorb energy before breaking. They focused on basalt, a common volcanic rock found in the underground caverns of southwestern China. By creating hundreds of computer models of fractured rock at different sizes, ranging from small five-meter cubes to massive fifty-meter blocks, they traced the path from a stable, elastic rock to a crumbling, failed one. Their work reveals that the rock can appear perfectly stable and representative at a certain size, yet still be hiding a dangerous tendency to fail much earlier and more violently than a smaller sample would suggest.

The researchers began by building a digital library of the rock's internal structure. They used real-world data from an exploration tunnel to create two types of fractured rock models: one representing "weakly weathered" rock, which has been exposed to air and water and contains more cracks, and another representing "fresh" rock, which is deeper and more intact. They generated two hundred and forty unique three-dimensional networks of fractures for each rock type, varying the size of the virtual cube from five meters to fifty meters. First, they tested how the rock deformed under pressure without actually breaking it. They found that the average stiffness of the rock, and the number of cracks per volume, became statistically stable once the cube reached ten meters in size. At this point, the average number of cracks and the surface area of the fractures stopped changing significantly as the sample grew larger.

However, stability in the number of cracks does not mean the sample is ready for engineering design. The researchers noted a crucial geometric constraint: the largest cracks in their models were thirty meters long. A ten-meter cube is simply too small to contain a thirty-meter crack without cutting it off at the edges. Therefore, they argued that a model must be at least thirty meters wide to be a "boundary-admissible" sample, meaning it can hold the full range of fracture sizes present in the real rock without artificial truncation. This established a new, larger baseline for what counts as a valid sample size, pushing the threshold from ten meters up to thirty.

With the geometric rules set, the team moved to the most critical part of the experiment: simulating how the rock actually breaks. They took one carefully calibrated model for each rock type and size, sliced it into two-dimensional sections, and subjected them to intense compression, mimicking the stress of an underground excavation. They watched the entire process, from the initial push to the moment the rock reached its peak strength and began to crumble. The results were surprising. While the stiffness of the rock changed very little as the models grew from five to fifty meters, the strength of the rock dropped dramatically. The peak stress the rock could withstand decreased by nearly thirty percent as the size increased. Even more striking was the energy the rock could absorb before failing. This energy, which represents the work the rock can do before breaking, plummeted by more than fifty percent in the largest models compared to the smallest ones.

The study explains this discrepancy by looking at how the rock fails. In the smaller models, the cracks were fewer and more isolated. As the models grew larger, they contained more long fractures and more opportunities for small cracks to link up. The researchers observed that in the larger models, local cracks would connect across the entire sample, forming a continuous path that caused the rock to collapse much sooner. This process, known as fracture localization, meant that the rock reached its breaking point at a much lower stress level, even though its initial stiffness remained the same. The researchers calculated that the loss of energy capacity was almost entirely due to this drop in strength, rather than a change in how stiff the rock was. They found that the relationship between strength and energy followed a precise mathematical rule where a small drop in strength leads to a massive drop in energy capacity, a fact that was confirmed by their simulations.

The team also examined the direction in which the rock was loaded. Even in the largest fifty-meter models, the rock did not behave the same way in every direction. Depending on the orientation of the cracks relative to the pressure, the rock could be significantly stronger or weaker. The average strength might have stabilized around forty meters, but the difference between the strongest and weakest directions remained substantial. This means that a model that looks "representative" on average could still be dangerously misleading if the actual engineering structure happens to align with the weakest direction. The study showed that the rock's internal structure remained anisotropic, or directionally dependent, even at the largest scales tested.

Ultimately, the research suggests that there is no single "magic number" for the size of a rock sample that works for all engineering purposes. A ten-meter sample might be enough to count the cracks, and a thirty-meter sample might be needed to include the full range of fracture sizes, but neither is sufficient to predict how the rock will fail under load. Crucially, the study found that even at the largest fifty-meter scale tested, the models failed to resolve the convergence of energy absorption or the behavior of the most conservative (weakest) loading directions. To understand the energy absorption and the risk of sudden collapse, engineers may need models even larger than fifty meters, or they must accept that their predictions will carry a significant margin of uncertainty. The study concludes that the stability of a rock's stiffness is a poor indicator of its stability in failure. A rock can look solid and predictable on a small scale, yet hide a complex network of connections that will cause it to shatter unexpectedly when the scale is large enough to let those connections form. This distinction is vital for the safety of underground structures, reminding engineers that the behavior of a rock mass is not just about how many cracks it has, but how those cracks are arranged and how they interact when the pressure is applied.