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We investigate the emergence of isotropic linear elasticity in amorphous andpolycrystalline solids, via extensive numerical simulations. We show that theelastic properties are correlated over a finite length scale $\xi_E$, so thatthe central limit theorem dictates the emergence of continuum linear isotropicelasticity on increasing the specimen size. The stiffness matrix of systems offinite size $L > \xi_E$ is obtained adding to that predicted by linearisotropic elasticity a random one of spectral norm $(L/\xi_E)^{-3/2}$, in threespatial dimensions. We further demonstrate that the elastic length scalecorresponds to that of structural correlations, which in polycrystals reflectthe typical size of the grain boundaries and length scales characterizingcorrelations in the stress field. We finally demonstrate that the elasticlength scale affects the decay of the anisotropic long-ranged correlations oflocally defined shear modulus and shear stress.

Emergence of linear isotropic elasticity in amorphous and polycrystalline materials