🔬 condensed matter

Response of a Model Glass to Athermal Quasistatic Pinching

This paper numerically demonstrates that athermal quasistatic pinching serves as a minimal local probe for amorphous solids, revealing a transition from linear elastic, dipolar responses to plastic regimes where the nature of rearrangements—ranging from localized events in well-annealed glasses to system-spanning cascades in poorly annealed ones—is governed by both the imposed extension and the material's stability.

Original authors: Takumi Nagasawa, Kirsten Martens, Jean-Louis Barrat, Misaki Ozawa

Published 2026-09-03

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CC BY 4.0

Original authors: Takumi Nagasawa, Kirsten Martens, Jean-Louis Barrat, Misaki Ozawa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world made of tiny, jumbled particles that have frozen in place, not because they cooled down into a crystal, but because they were trapped in a chaotic, disordered state. This is the nature of amorphous solids, a category of materials that includes everything from window glass and metallic alloys to the sand in a dune and the cells in your body. Unlike a crystal, where atoms sit in a perfect, repeating grid, these materials are a mess of irregular packing. Yet, despite their internal disorder, they behave remarkably similarly when you push or pull on them. They are solid, they resist deformation, but if you push hard enough, they eventually give way, flowing like a thick liquid. Understanding exactly how these materials transition from rigid to flowing is one of the great challenges in physics, because it governs why a bridge might hold or why a glass might shatter.

For decades, scientists have studied how these materials react to global forces, such as stretching a whole block of metal or squeezing a container of sand. But a more subtle question has remained: what happens when you poke just two tiny particles inside the material? If you were to grab two neighboring atoms and pull them apart, how would the rest of the material respond? Would the disturbance stay local, or would it ripple out to shake the entire structure? This question is central to understanding the fundamental building blocks of how solids break and flow. A new study by researchers at the University of Grenoble Alpes in France has tackled this by simulating a process they call "pinching," where they virtually grab two particles and slowly pull them apart to see what happens.

The researchers built a digital model of a glassy material, filling a square box with thousands of tiny spheres of three different sizes. They created many versions of this digital glass, some that were very stable and others that were less stable, mimicking the difference between a carefully cooled glass and one that was cooled too quickly. They then selected a random pair of touching particles and applied a force to pull them apart, increasing the distance between them in tiny, controlled steps. This process was done without heat, meaning the material was allowed to settle into its most comfortable position after every tiny movement, allowing the scientists to watch the material's reaction with perfect clarity.

When the researchers pulled the particles apart just a little bit, the material behaved exactly as classical physics predicts for a solid. The two particles moved apart, and the surrounding particles shifted slightly to accommodate the change, creating a pattern of movement that spread out in a specific, four-lobed shape. This pattern, which looks like a four-leaf clover, is what physicists expect when a solid is deformed locally. The researchers found that this simple, predictable behavior held true even when they looked at distances just a few particle diameters away from the pinch. It was a confirmation that the standard rules of elasticity, which usually apply to large, smooth objects, work perfectly well even at the microscopic scale of individual atoms, provided the material is stable and the pull is gentle.

However, the story changed when the researchers pulled the particles further apart. At a certain point, the local cage of particles holding the two pulled-apart atoms together finally broke. This was the moment the material stopped acting like a spring and started acting like a plastic substance. In the most stable glasses, this breaking was a contained event. The particles rearranged themselves right where the pinch happened, and the disturbance died out quickly. The rest of the material remained calm, and the event was a localized, isolated rearrangement.

But in the less stable, poorly prepared glasses, the story was very different. When the cage broke in these materials, it did not stay quiet. The initial rearrangement sent out a shockwave of stress that triggered other particles far away to move as well. One small slip caused another, which caused another, leading to a chain reaction. In these unstable materials, a single pinch could set off a cascade of movements that spread across the entire system, turning a tiny local event into a massive, system-wide avalanche. The researchers observed that the more unstable the glass was, the more likely it was for a small, local pull to trigger a large-scale collapse.

The team also discovered that the way the material responded depended heavily on how it was prepared. The stable glasses, which were cooled slowly in the simulation, resisted these cascades and kept their plastic events small and contained. The unstable glasses, which were cooled rapidly, were prone to these widespread avalanches. This suggests that the history of how a material is made—how it was cooled or prepared—leaves a lasting mark on how it reacts to stress. A well-prepared glass is like a disciplined crowd that absorbs a push without panicking, while a poorly prepared one is like a crowd on the edge of a stampede, where a single push can send everyone running.

By mapping out exactly how the particles moved and how the energy was released, the researchers were able to connect these microscopic events to the broader theory of how solids deform. They found that even when a plastic event occurred, the way the material moved away from the event still followed the same mathematical patterns as the initial elastic response, just with a larger magnitude. This means that the fundamental language of how these materials move is consistent, whether they are bending slightly or breaking apart. The key difference is not in the language of the movement, but in the stability of the material itself, which determines whether that movement stays small or grows into a catastrophe.

This work provides a clear, detailed picture of the smallest possible unit of deformation in a solid. It shows that the transition from a solid that springs back to a material that flows is not a sudden, mysterious jump, but a process governed by the stability of the material and the size of the disturbance. The study confirms that while the rules of elasticity hold true even at the smallest scales, the potential for chaos lies just beneath the surface, waiting to be triggered by the right combination of instability and force. For scientists trying to design stronger materials or understand how complex systems fail, this offers a new way to look at the world: by watching how a single pair of particles reacts to a pinch, one can understand the fate of the entire structure.

Technical Summary: Response of a Model Glass to Athermal Quasistatic Pinching

Problem Statement
Amorphous solids, despite variations in particle size and microscopic interactions, exhibit universal mechanical behaviors under external loading. While global deformation modes like shear, tension, and compression are well-studied, the response to localized force dipoles—specifically a "pinching" protocol where two contacting particles are pulled apart—remains less explored, particularly regarding the transition from elastic to plastic regimes. Furthermore, the dependence of this response on the stability of the glass (determined by preparation protocols) and the potential for a local pinching event to trigger distant, system-spanning plastic rearrangements (avalanches) are not fully understood. This study aims to bridge the gap between linear elasticity theory and numerical simulations to characterize the mechanical response of model glasses to pinching across varying degrees of stability.

Methodology
The authors employ athermal quasistatic (AQS) simulations on a two-dimensional ternary mixture of Lennard-Jones particles (small, medium, and large sizes) designed to prevent crystallization.

Glass Preparation: Samples are generated by rapidly quenching equilibrated supercooled liquids prepared at different initial temperatures ( $T_{ini}$ ). This yields three distinct stability classes: stable glasses ( $T_{ini} = 0.25$ ), moderately annealed glasses ( $T_{ini} = 0.35$ ), and poorly annealed glasses ( $T_{ini} = 1.0$ ). System sizes range from $N=1000$ to $N=64000$ .
Pinching Protocol: Starting from inherent structures at zero temperature, a random pair of neighboring particles is selected. An external harmonic potential is added to the system energy, controlling the separation distance $\ell$ between the two particles. The separation is increased incrementally ( $\Delta \ell = \ell - \ell_0$ ) in a length-controlled manner, with energy minimization (using the FIRE algorithm) performed after each step.
Analysis: The study compares simulation results against predictions from linear elasticity theory. Key metrics include the displacement field $\mathbf{u}(\mathbf{r})$ , the mean radial decay of displacement magnitude $u(r)$ , the probability distribution of displacement magnitudes $P(u)$ , and the non-affine displacement measure $D^2_{min}$ to quantify plastic activity. A specific method is introduced to extract the principal axis of plastic deformation to correctly average displacement fields in the plastic regime.

Key Contributions and Results

Theoretical Framework:
The authors establish a theoretical connection between pinching (modeled as a single force dipole) and shear transformations (modeled as a pair of perpendicular force dipoles). Using linear elasticity, they derive the displacement fields for both cases. They demonstrate that in the incompressible limit ( $\nu \to 1$ ), the displacement field generated by a pinching event is exactly half that of a "pinching-contraction" event (equivalent to a shear transformation). For compressible materials, the response retains a quadrupolar-like symmetry but with lobe intensities sensitive to the Poisson ratio.
Elastic Regime Response ( $\Delta \ell \approx 0.5$ ):
- Validation of Continuum Theory: For small extensions, the system responds elastically. The averaged displacement fields exhibit a four-lobed, quadrupolar-like pattern.
- Quantitative Agreement: There is remarkable quantitative agreement between the simulation results and linear elasticity predictions, even at distances as small as a few particle diameters. This confirms that continuum elasticity accurately describes the response to highly localized perturbations in stable and moderately annealed glasses.
- Scaling Laws: The mean displacement magnitude decays as $u(r) \sim r^{-1}$ , and the probability distribution of displacement magnitudes follows $P(u) \sim u^{-3}$ , consistent with 2D linear elasticity. These scaling regimes are robust across system sizes for stable glasses.
- Stability Dependence: In poorly annealed glasses, the elastic signal is more diffuse due to inelastic effects and the triggering of additional plastic rearrangements even at small extensions.
Plastic Regime Response ( $\Delta \ell \approx 2.0$ ):
- Transition to Plasticity: As the imposed extension increases, the local cage structure breaks, triggering a plastic event.
- Localized vs. Delocalized Response: The nature of the plastic response depends critically on glass stability.
  - Stable Glasses: Plastic events remain localized around the pinched particles. The resulting displacement field remains well-described by linear elasticity (Eshelby-like), with the plastic rearrangement acting as a local eigenstrain.
  - Poorly Annealed Glasses: The primary plastic event triggers a cascade of secondary plastic rearrangements in distant soft regions. This leads to a delocalized, avalanche-like response that deviates significantly from linear elasticity predictions at large distances.
- Principal Axis Analysis: In the plastic regime, the orientation of the quadrupolar pattern fluctuates. By extracting the principal axis of deformation for each realization, the authors show that the averaged response in stable glasses still aligns with linear elasticity, whereas poorly annealed glasses show isotropic, delocalized spreading of plastic activity.
Control Parameter Dependence:
The study demonstrates that the imposed extension $\Delta \ell$ serves as a control parameter to continuously tune the system from the elastic to the plastic regime. The transition is marked by discontinuities in potential energy and spring energy, corresponding to the release of accumulated elastic stress.

Significance and Claims
The paper claims that pinching provides a "minimal local probe" of amorphous solids. The results demonstrate that:

Linear elasticity remains a valid and accurate description of the mechanical response down to the particle scale, provided the system is sufficiently stable and the perturbation is not large enough to trigger widespread plasticity.
The stability of the glass dictates the spatial extent of plasticity: stable glasses exhibit localized plastic events, while poorly annealed systems exhibit collective, system-spanning plasticity triggered by a single local event.
The pinching protocol offers an alternative mechanism to drive amorphous solids, distinct from global shear or thermal activation. This has potential implications for understanding elementary processes in glassy dynamics, such as localized excitations and structural changes induced by X-ray irradiation, where local cage-breaking events act as sources of mechanical perturbation.

The authors maintain a modest stance, noting that while the connection between pinching and shear transformations is established theoretically, the specific role of "four-particle key cores" (recently identified as elementary defects) in these processes remains a subject for future investigation. The study focuses on elucidating the interplay between elasticity, elementary rearrangements, and the emergence of collective plasticity under localized forcing.

Technical Summary: Response of a Model Glass to Athermal Quasistatic Pinching

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