The influence of nanoparticles on the macroscopic rigidity of amorphous solids

In the random matrix model with translational symmetry, it is studied the influence of nanoparticles on the macroscopic rigidity of an amorphous system. The numerical analysis shows that the macroscopic theory of elasticity can be applied if the radius of nanoinclusions R is big enough. In this case, it gives an additional contribution to the Young's modulus $\Delta E\sim R^{3}$. However, as the radius of nanoinclusions decreases, this dependence becomes quadratic, $\Delta E\sim R^{2}$. Theoretical result for Young's modulus was obtained by reducing the energy of the whole system to a sum of quadratic forms and by applying the Gauss–Markov theorem. From this theorem, it follows that the stiffness of the system depends on the difference between the number of bonds and the number of degrees of freedom, which is proportional to the surface area of nanoinclusions. It is shown, that there is a scale of nanoinclusion radius, which characterizes a scale of inhomogeneity of amorphous solids. It determines the smallest characteristic size of nanoinclusions, at which the macroscopic theory of elasticity can be applied.