Approximate analytical description of the nonaffine response of amorphous solids

An approximation scheme for model disordered solids is proposed that leads to the fully analytical evaluation of the elastic constants under explicit account of the inhomogeneity (nonaffinity) of the atomic displacements. The theory is in quantitative agreement with simulations for central-force systems and predicts the vanishing of the shear modulus at the isostatic point with the linear law $\mu ~(z-2d)$, where $z$ is the coordination number. The vanishing of rigidity at the isostatic point is shown to be a consequence of the canceling out of positive affine and negative nonaffine terms.

Figure 1

The theoretical predictions for the shear modulus, Eq. (17), and the simulation results of Ref. 12. No fitting parameter is used in the comparison. $\kappa =1,R_0=1$.

Figure 2

1 (red), The DOS calculated for a system of $N=10\hspace{0.5em}000$ particles according to Eq. (12) (solid line) and $z=8$. 2 (blue), Simulations for repulsive unstressed harmonic packings of Ref. 17 with $\varphi -{\varphi }_c=0.1$, which corresponds to $z-6\simeq 2$.