The Influence of Nanoparticles on the Macroscopic Stiffness of Amorphous Solids

The influence of nanoinclusions on the macroscopic stiffness of amorphous systems was studied in the context of the random matrix model with translation symmetry. The numerical analysis of nanoinclusions, whose radius R is large enough, admits the use of the macroscopic theory of elasticity, defining the addition to the Young’s modulus as Δ E ~ R 3 . Nevertheless, a decrease in nanoinclusion radius makes this dependence quadratic, i.e., Δ E ~ R 2 . Reducing the energy of the whole system to a sum of quadratic forms enables the Young’s modulus to be evaluated via the Gauss—Markov theorem. As follows, the stiffness of a medium depends on the difference between the number of bonds and the number of degrees of freedom of a system, which is proportional to the nanoparticle surface area. Furthermore, the scale of heterogeneity of the amorphous solids corresponds to a certain nanoinclusion radius, which determines the lowest characteristic nanoparticle size and the applicability of the macroscopic theory of elasticity.