The Suspense in Failure: A Simple Model of Breakage Goes Universal

It’s a classic scene in action movies: The hero is dangling from a rope, staring down at certain death. Just as he starts climbing, a fiber snaps above his head. A suspenseful score swells as a hidden clock begins to count down until the final fiber breaks. We see another snap, and then another. Just in the nick of time, the hero lands safely on a ledge as the rope plummets into the depths.

“Neglecting a few ‘Deus ex machina‘ elements, the mechanical behavior of the rope is qualitatively well reproduced in action films,” says Jordi Baró, a researcher in Jörn Davidsen’s group at the University of Calgary.

The movies may get it right, but that doesn’t mean the physics is straightforward. In new research published in in the American Physical Society’s journal Physical Review E, Baró and Davidsen explored the physics behind this “ticking clock effect” and found a surprising connection to the failure of other types of materials.

The research didn’t start out as a tribute to action movies. Initially, Baró was studying the mechanical failure of porous materials like rocks and concrete. If you zoom in on the weak point of a concrete bridge or geological fault just before it fails, you’ll see a whole bunch of tiny fractures, called microcracks. By understanding microcracks and their role in mechanical failure, we can better predict and prevent the sometimes-disastrous consequences of such failures.

The goal of Baró’s work was to create a mathematical model that described the properties of microcracks observed in experiments. Common models of fracturing in solids provided a good starting place, but the results didn’t match up. In Baró’s words, “The models expected an increase in the size of microcracks as the material approach failure, but the experimental evidence was showing a stationary distribution of crack sizes.” Something was missing.

To get to the missing element, let’s revisit the action hero scenario and consider this question: When an action hero hangs from a rope, why don’t all of the fibers snap at once?

If you examined the fibers of a rope under a microscope, you’d see that all of them have defects. Defects influence the breaking point of a fiber, so different fibers snap under different amounts of tension. In general, the number of broken fibers is determined by the weight of the object. More rope fibers would snap under the weight of Arnold Schwarzenegger (250 pounds) than Jackie Chan (140 pounds).

This partially answers the question. It explains why fibers have different breaking points, but it doesn’t explain why, under the same weight, some fibers break sooner than others. It doesn’t explain the “ticking clock” effect that adds so much drama to these scenes.

Our action hero has a few tense seconds to get to safety because the fibers in the rope respond to the stress of an action hero’s weight by slowly deforming over time, stretching more and more until they finally snap. However, the rate at which they stretch slows down over time—a behavior called viscoelasticity, because it combines elements of viscous materials and elastic materials. Without viscoelasticity, all of the fibers that that would snap under the weight of that hero would snap immediately.

On examining the microcrack situation, Baró realized that the inconsistencies between the models and experimental results might disappear if you include a factor analogous to viscoelasticity. By running simulations of microcrack formations with and without the viscoelasticity factor, the team demonstrated that the viscoelastic model more closely matches real microcrack formation. This research marks the first time scientists have incorporated this kind of factor when modeling the properties of microcracks and how they evolve in time.

However, this progress is not the end of the story. The researchers also showed that in its simplest form, their mathematical expression provides a unified solution for a broad collection of fracture phenomena. This simple model explains the properties of fractures in solid materials across a wide range of scales and scenarios. For example, the ticking clock effect has analogies in the behavior of stressed crystals and glasses, the ultrasonic crackling noise that precedes the failure of human-made structures, and in the aftershocks at the scale of plate tectonics.

“The reduction of hours of work and pages of equations into a simple mathematical expression is always rewarding, even more so in this case” says Baró. He continues, “When solving conceptual models, preciseness gives certainty, but simplicity gives meaning to the result.” The new expression describes many different models regardless of their details—a property called universality—and that’s a holy grail in this kind of theoretical research.

When he’s not working, Baró enjoys a good action flick. “In my opinion the most important elements for a good action movie are a well-paced and well-timed narrative, a witty script and, of course, the presence of a big star,” he says. From now on, Baró will probably also have a special appreciation for action movies featuring a narrow escape from the abyss.

—Kendra Redmond

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