Particle displacements in the elastic deformation of amorphous materials: local fluctuations versus non-affine field

We study the local disorder in the deformation of amorphous materials by decomposing the particle displacements into a continuous, inhomogeneous field and the corresponding fluctuations. We compare these fields to the commonly used non-affine displacements, in an elastically deformed 2D Lennard-Jones glass. Unlike the non-affine field, the fluctuations are very localized, and exhibit a much smaller (and system size independent) correlation length, on the order of a particle diameter, supporting the applicability of the notion of local “defects” to such materials. We propose a scalar “noise” field to characterize the fluctuations, as an additional field for extended continuum models, e.g., to describe the localized irreversible events observed during plastic deformation. Introduction. – The nature of fluctuations in glasses and other amorphous materials out of equilibrium is of much current interest. While elasticity and plasticity are often employed for describing both crystalline and amorphous materials, their microscopic basis is well-established only in crystalline (or polycrystalline) materials, and relies on the periodicity of the microscopic structure (possibly with localized defects) [1, 2]. As in crystal plasticity, localized rearrangements appear to play an important role in the plastic deformation of amorphous materials [3–5], but the lack of underlying order renders the identifications of localized “defects” difficult. In crystals (with a simple unit cell) under homogeneous deformation, the particle displacements conform to the imposed (affine) strain, but in amorphous materials they do not [6]. The non-affine displacements (obtained by subtracting the expected homogeneous deformation) have recently been studied in experiments and simulations of different amorphous systems (e.g., glasses, colloids, granular materials and foams [7–14]). They are typically of the same order of magnitude as the relative affine displacements of neighbor(a) E-mail: [email protected] (b)E-mail: [email protected] (c)E-mail: [email protected] ing particles, and therefore cannot be considered a small correction: ignoring them, or treating them as a perturbation, yields highly inaccurate estimates for macroscopic material properties such as the elastic moduli [8, 15, 16]. Considering the non-affine displacements as a fluctuation, or “noise” [8, 10], poses difficulties since they exhibit long range correlations [8, 10–13, 17] which would render the contribution of such “noise” dominant at large scale. In this Letter we show, using numerical simulations of a two dimensional (2D) Lennard-Jones glass subject to small elastic deformation, that the main features of the non-affine field can be captured by a continuous, inhomogeneous (subsystem scale) displacement field. However, the microscopic displacements exhibit significant fluctuations with respect to this field. A full characterization of the displacements therefore requires not only a distinction between an affine and non-affine contribution, but also between a continuous and a fluctuating part. We present the first study of the local fluctuations, and show that their properties are very different from those of the non-affine field: they are essentially uncorrelated and extremely localized; their distribution is qualitatively different. Furthermore, unlike the non-affine displacement, whose correlation depends on the system size (see [13] and below),