Emergence of linear isotropic elasticity in amorphous and polycrystalline materials

Abstract

We investigate the emergence of isotropic linear elasticity in amorphous and polycrystalline solids, via extensive numerical simulations. We show that the elastic properties are correlated over a finite length scale ξE, so that central limit theorem dictates the emergence of continuum linear isotropic elasticity on increasing the specimen size. The stiffness matrix of systems of finite size L > ξE is obtained adding to that predicted by linear isotropic elasticity a random one of spectral norm (L/ξE) −3/2 , in three spatial dimensions. We further demonstrate that the elastic length scale corresponds to that of structural correlations, which in polycrystals reflect the typical size of the grain boundaries and length scales characterizing correlations in the stress field. We finally demonstrate that the elastic length scale affects the decay of the anisotropic long-ranged correlations of locally defined shear modulus and shear stress.

Unifying description of the vibrational anomalies of amorphous materials

“…We have determined FET's disorder parameter, γ, and the elastic correlation length, ξ e , investigating the size dependence of the fluctuations of the elastic response. We characterize the elastic response of our samples evaluating [42,43] their stiffness matrix, Ĉ. We determine this matrix by measuring the stress changes resulting from six independent strain deformations followed by energy minimization in the linear response regime.…”

“…We study the fluctuations in the shear response [43] investigating the size dependence of the normalized fluctuations…”

“…To validate this theory, we determine ξ e and γ investigating the system and coarse-grained size dependence of the elastic properties, finding consistent results. We follow standard approaches [39,42,43] and provide details in the Supplementary Material [44]. We find that a single length scale characterizes the dependence the shear modulus fluctuations on the system size, so that γ is a non-dimensional measure of the correlation volume, γ ∝ ξ 3 e .…”