🔬 condensed matter

Symmetry Breaking and Energy Dissipation in the Mechanical Response of Amorphous Solids

This paper investigates the link between energy dissipation and symmetry breaking in amorphous solids under quasi-static strain, demonstrating through a solvable mesoscopic theory and numerical simulations that dissipation arises specifically from rotational symmetry breaking during a sharp transition between conservative and dissipative mechanical responses.

Original authors: Itamar Procaccia, Tuhin Samanta

Published 2026-08-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Itamar Procaccia, Tuhin Samanta

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

The Secret Life of Jiggly Jellies

Imagine you are a materials scientist, but instead of studying steel beams or glass windows, you are obsessed with the weird, wobbly stuff in between: amorphous solids. Think of glass, plastic, or even a jar of jam. These materials look solid, but unlike a perfect crystal (where atoms are lined up in neat, repeating rows like soldiers), their insides are a chaotic mess, like a crowd of people at a concert who can't find their seats. Scientists have long been puzzled by what happens when you squeeze and stretch these messy materials over and over again.

When you push on a material, it usually pushes back. If you let go, it snaps back to its original shape, and all the energy you put in comes back out. This is called being "conservative." But sometimes, materials get tired. They lose energy as heat, they shift their internal structure, and they don't quite return to where they started. This loss of energy is called "dissipation." In the world of physics, there's a famous rule (Noether's theorem) that says if energy isn't conserved, something "symmetry" must have broken. Symmetry is just a fancy word for balance or perfect repetition. If you spin a perfect circle, it looks the same from every angle (rotational symmetry). If you slide a perfect pattern, it looks the same (translational symmetry). The big question for scientists has been: Which specific kind of balance is breaking to cause that energy loss in these messy solids? Understanding this helps engineers design better materials that don't wear out from fatigue, like the parts in a car engine or a bridge that sways in the wind.

The Great Inflation Experiment

In this study, the researchers decided to strip away all the complexity and look at just one simple "breathing" cycle of a messy material to see where the energy goes. They imagined a ring of tiny, frictionless disks (like a pile of coins) trapped between two walls. In the middle, they placed one special, giant disk. The experiment was a simple three-step dance:

The Start: The system sits quietly, perfectly balanced.
The Inflate: The giant center disk suddenly swells up, pushing the surrounding disks outward. The system settles down into a new, squeezed shape.
The Deflate: The giant center disk shrinks back to its original size. The system settles down again.

If the material were a perfect, bouncy rubber ball, it would return to its exact starting energy after the center disk shrank. But the researchers found that depending on how tightly the disks were packed (the pressure), the outcome was wildly different.

When the disks were packed very tightly (high pressure), the material acted like a perfect spring. The energy put in to inflate the center came right back out when it deflated. Nothing was lost, and the material's internal balance remained unbroken. It was a boring, but perfect, elastic cycle.

However, when the pressure was lower, the story changed dramatically. As the center disk swelled, the surrounding disks didn't just move smoothly; they tumbled, slid, and rearranged themselves in chaotic little avalanches. When the center disk shrank back, the material didn't return to its original state. Energy was lost. The researchers used computer simulations to track exactly what happened inside this mess. They found that the chaotic rearrangements created "topological charges"—think of them as invisible swirls or vortices in the way the disks moved.

The big discovery was identifying which symmetry broke to cause this energy loss. The researchers had two suspects:

Translation Symmetry Breaking: This would mean the material lost its ability to look the same if you slid it sideways.
Rotational Symmetry Breaking: This would mean the material lost its ability to look the same if you spun it around.

By carefully measuring the "screening parameters" (which are like mathematical fingerprints of how the material resists movement), the team found that the energy loss was directly tied to the rotational symmetry breaking. In simple terms, the chaotic avalanches caused the internal "dipole fields" (the way the disks pushed on each other) to twist and rotate relative to the direction they were moving. This twist meant that the force pushing the material out wasn't perfectly aligned with the force pulling it back in. It was like trying to push a door open while someone else is pulling it slightly sideways; you lose energy fighting that misalignment.

The paper explicitly rules out the idea that simple sliding (translational symmetry breaking) was the main culprit for the energy loss in this specific setup. Instead, the simulations showed a clear, linear relationship: the more the rotational symmetry was broken (the more the internal forces twisted), the more energy was dissipated. The researchers didn't just guess this; they ran thousands of simulations with different pressures and inflation sizes, and the math held up every time. They even derived a specific equation showing that the total energy lost is directly proportional to this "twist" factor.

So, the next time you wonder why a piece of plastic or glass eventually cracks or heats up after being bent back and forth, remember this: it's not just about the material getting tired. It's about the microscopic chaos inside losing its perfect spin, twisting the forces just enough to turn your hard work into wasted heat.

Technical Summary: Symmetry Breaking and Energy Dissipation in the Mechanical Response of Amorphous Solids

Problem Statement
The precise physical origin of energy dissipation in amorphous solids subjected to cyclic strain protocols remains elusive, despite extensive study of phenomena such as fatigue, self-organization, and yielding. While Noether's theorem establishes that a loss of energy conservation is associated with continuous symmetry breaking, the specific symmetry responsible for dissipation in quasi-static cyclic driving of amorphous solids has not been identified. This paper addresses the question of which symmetry breaking mechanism drives dissipation during a single cycle of strain, particularly in regimes where time-dependent effects are absent.

Methodology
The authors investigate a model system consisting of an assembly of frictionless, bi-dispersed disks confined in a two-dimensional annulus (between an inner radius $r_{in}$ and outer radius $r_{out}$ ). The system is prepared in mechanical equilibrium with a packing fraction above the jamming threshold. The study employs a specific cyclic protocol:

Initial Equilibration: The system is relaxed to a stable state with energy $E_b$ and pressure $P_b$ .
Inflation: The central disk (acting as the inner boundary) is inflated by a percentage (5% to 25%), and the system is re-equilibrated.
Deflation: The central disk is returned to its original radius, and the system is re-equilibrated to a final state with energy $E_f$ and pressure $P_f$ .

The study combines numerical simulations (using LAMMPS with Hertzian contact forces) and mesoscopic analytical theory. The simulations track displacement fields and energy changes ( $\Delta E = E_b - E_f$ ) across various pressures and inflation amplitudes. Theoretical analysis utilizes continuum mechanics equations to describe the displacement field, distinguishing between regimes governed by classical elasticity and those involving plastic rearrangements.

Key Results
The simulations reveal a sharp transition in mechanical behavior dependent on the initial pressure $P_b$ relative to a critical pressure $P_c$ :

High-Pressure Regime ( $P_b > P_c$ ): The response is quasi-elastic and conservative. The displacement field after deflation returns to zero, and energy dissipation is negligible ( $\Delta E/E_b \approx O(10^{-5})$ ). The system obeys the classical elasticity equation $\Delta \mathbf{d} + (\tilde{\lambda} + 1)\nabla(\nabla \cdot \mathbf{d}) = 0$ , and no symmetry breaking occurs.
Low-Pressure Regime ( $P_b < P_c$ ): The response becomes dissipative ( $\Delta E \neq 0$ ). Plastic avalanches generate topological charges (quadrupolar and dipolar), leading to a modified displacement equation:
$\Delta \mathbf{d} + (\tilde{\lambda} + 1)\nabla(\nabla \cdot \mathbf{d}) + \mathbf{\Gamma}\mathbf{d} = 0$
Here, $\mathbf{\Gamma}$ is a screening tensor containing diagonal terms (associated with translational symmetry breaking) and off-diagonal terms (associated with rotational symmetry breaking).

Mechanism of Dissipation
The paper identifies rotational symmetry breaking as the fundamental source of energy dissipation.

The off-diagonal screening parameter $\kappa_o$ (the "odd" part of the tensor) represents a rotation of the local dipole field relative to the local displacement field.
When $\kappa_o = 0$ , the system is conservative. When $\kappa_o \neq 0$ , the system exhibits chiral symmetry breaking.
The total energy dissipation is derived as $\Delta E = \kappa_o^2 I$ , where $I$ is an integral of the displacement field. This demonstrates that dissipation arises specifically from the non-zero off-diagonal components of the screening tensor, which break rotational symmetry.
Numerical data confirms a linear relationship between the normalized energy dissipation and the odd screening parameter $\kappa_o$ , validating the theoretical prediction that dissipation vanishes when rotational symmetry is preserved.

Significance and Claims
The authors propose that analyzing a single strain cycle provides a clear window into the physics of dissipation in amorphous solids. The primary contribution of this work is the identification of rotational symmetry breaking (manifested via the odd screening parameter $\kappa_o$ ) as the specific mechanism responsible for energy loss in quasi-static cyclic loading below a critical pressure.

The paper posits that this simple annular system serves as an analog to the harmonic oscillator in molecular physics: an analytically tractable model that reveals complex physics, including screening, symmetry breaking, and phase transitions. The work clarifies that while both translational and rotational symmetries are broken in the dissipative regime, it is specifically the breaking of rotational symmetry that leads to the non-conservation of energy.

The Secret Life of Jiggly Jellies

The Great Inflation Experiment

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