Nonlinear visco-elasto-plastic model for the stress-strain and stress-relaxation in metallic glass

Imagine a material that looks like a solid metal but is built like a liquid, with its atoms frozen in a disordered jumble rather than arranged in neat, repeating crystals. This is a metallic glass, a substance known for being incredibly strong yet prone to sudden, brittle failure. To understand how these materials bend, break, or flow, scientists have long relied on a classic picture: the material is caught in a tug-of-war between two forces. On one side is elasticity, the tendency to snap back like a rubber band when stretched. On the other is relaxation, the slow, internal rearrangement of atoms that allows the material to flow like a thick syrup. For decades, researchers have measured how these materials relax over time in separate experiments, and how they stretch and break in others, but the two sets of data rarely spoke to each other in a single, unified story.

A team of researchers has now bridged that gap by developing a new way to describe the behavior of metallic glasses. They created a model that connects the slow, internal clock of the material's relaxation directly to the rapid, violent events of stretching and breaking. By treating the metal as a combination of a spring and a dashpot—a device that resists motion like a piston in thick oil—they were able to map out exactly how the material transitions from a rigid solid to a flowing liquid under stress. Their work reveals that the key to predicting when a metallic glass will yield or snap lies in understanding how its internal relaxation speed changes as it is deformed.

The researchers focused on a specific type of metallic glass made from zirconium, titanium, copper, nickel, and beryllium, known as Vitreloy 1. They began by looking at how this material relaxes when held at a constant stretch. In a typical experiment, if you pull a piece of metal and hold it, the stress inside it slowly fades as the atoms shift into more comfortable positions. For this glass, that fading does not happen at a steady, predictable rate. Instead, it follows a complex pattern where the relaxation stretches out over a wide range of times, a behavior that is characteristic of disordered, glassy states. The team measured this relaxation at different temperatures, finding that as the material got warmer, the atoms moved much faster, with the time it took for the stress to fade dropping from 22 seconds at one temperature to just 2.7 seconds at a slightly higher one.

With this baseline of how the material relaxes in a calm state, the team then turned to the chaotic world of stretching the metal at a constant speed. When you pull a metallic glass quickly, the stress rises sharply, reaches a peak, and then drops before settling into a steady flow. This peak, known as a stress overshoot, is a critical moment where the material begins to yield and deform permanently. The researchers found that the height of this peak and the point at which it occurs depend heavily on how fast you pull and how hot the material is. If you pull slowly or keep the material warm, the atoms have time to rearrange themselves as you stretch, so the stress peak is lower and the material flows more easily. If you pull fast or keep it cool, the atoms cannot keep up, the stress builds to a higher peak, and the material is more likely to snap.

To explain this, the team introduced a concept called "material time." Think of this not as the seconds ticking on a clock, but as an internal clock that measures how much the material has actually changed or relaxed. In a simple, calm situation, this internal clock ticks at the same rate as the real-world clock. However, as the material is stretched into the nonlinear regime where it starts to yield, this internal clock speeds up or slows down relative to the real world. The researchers calculated this change by comparing the material's behavior during stretching to its known relaxation behavior. They found that at the very beginning of stretching, the internal clock matches the real clock, meaning the material is behaving like a simple elastic solid. But as the stretching continues and the material begins to yield, the internal clock diverges, signaling that the atoms are rearranging much faster than they would in a calm state.

This divergence is quantified by a ratio that tells the researchers how much the material's internal dynamics have changed due to the stress. When the material is first stretched, this ratio is close to one, indicating that the material is still in its familiar, linear state. As the stress builds and the material yields, the ratio grows, showing that the deformation is now driving the atoms to rearrange in a way that is distinct from their natural relaxation. This shift marks the transition from the material acting like a solid to acting like a flowing liquid. The team used this insight to successfully predict the entire stress-strain curve, from the initial stiff rise to the peak overshoot and the subsequent softening, across different temperatures and pulling speeds.

The study also allowed the researchers to estimate the size of the tiny clusters of atoms that move together when the material yields. By analyzing how the internal clock changed with strain, they calculated that these moving clusters occupy a volume of roughly 90 to 97 cubic angstroms. This number provides a physical sense of the scale at which the material fails, suggesting that the deformation is not happening atom by atom, but in small, coordinated groups. The findings confirm that the competition between the speed of the applied stretch and the speed of the material's internal relaxation is the fundamental driver of its mechanical behavior.

Ultimately, this work provides a simple but powerful connection between two previously separate views of metallic glasses. It shows that the complex, nonlinear way these materials break and flow can be understood by looking at how their internal relaxation time changes under pressure. The model successfully captures the initial elastic response, the dramatic stress overshoot, and the steady flow that follows, all by linking the material's history of relaxation to its current state of deformation. While the current model focuses on this specific zirconium-based glass, the researchers plan to extend this approach to other amorphous materials, such as polymer glasses, to see if the same principles of internal time and relaxation govern their behavior as well.