Continuum limit of amorphous elastic bodies (III): Three dimensional systems

TL;DR

This study extends previous work to three-dimensional amorphous solids, analyzing how classical elasticity breaks down at a specific length scale and linking vibrational anomalies to material inhomogeneities.

Contribution

It provides the first systematic analysis of the continuum limit in 3D amorphous systems, identifying a characteristic length scale where elasticity theory fails.

Findings

Elasticity breaks down below length scale ~23 molecular sizes.

Non-affine displacement correlations are characterized.

The Boson peak is linked to inhomogeneities at sub-ξ wavelengths.

Abstract

Extending recent numerical studies on two dimensional amorphous bodies, we characterize the approach of elastic continuum limit in three dimensional (weakly polydisperse) Lennard-Jones systems. While performing a systematic finite-size analysis (for two different quench protocols) we investigate the non-affine displacement field under external strain, the linear response to an external delta force and the low-frequency harmonic eigenmodes and their density distribution. Qualitatively similar behavior is found as in two dimensions. We demonstrate that the classical elasticity description breaks down below an intermediate length scale $\xi$, which in our system is approximately 23 molecular sizes. This length characterizes the correlations of the non-affine displacement field, the self-averaging of external noise with distance from the source and gives the lower wave length bound for the applicability of the classical eigenfrequency calculations. We trace back the "Boson-peak" of the density of eigenfrequencies (obtained from the velocity auto-correlation function) to the inhomogeneities on wave lengths smaller than $\xi$.