Do athermal amorphous solids exist?

We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist, one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of nonaffine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient $B_2$ has anomalous fluctuations and the second nonlinear coefficient $B_3$ and all the higher order coefficients (which are nonzero by symmetry) diverge in the thermodynamic limit. These results call into question the existence of elasticity (or solidity) of amorphous solids at finite strains, even at zero temperature. We discuss the physical meaning of these results and propose that in these systems elasticity can never be decoupled from plasticity: the nonlinear response must be very substantially plastic.

Figure 1

The nonaffine velocities $\mathit{v}\equiv \frac{d\mathit{u}}{d\gamma }=-{\mathit{H}}^{-1}·\mathit{\Xi }$ for a typical quenched realization of a two-dimensional glass forming model, with $N=10\hspace{0.5em}000$.

Figure 2

Cartoon of a typical stress vs strain curve of a single instance of the numerical experiment performed in Ref. 6. Each undeformed system was strained until the first mechanical instability was encountered at some strain value ${\gamma }_P$. Statistics of $\Delta {\gamma }_{\mathrm{iso}}\equiv {\gamma }_p$ were collected for a variety of system sizes, in both two and three dimensions.

Figure 3

The mean strain interval before the first mechanical instability $\langle \Delta {\gamma }_{\mathrm{iso}}\rangle$, for two dimensions (left) and three dimensions (right). The continuous lines represent the scaling law Eq. (26).

Figure 4

An example of a single measurement of $D_2$ close to a mechanical instability at ${\gamma }_P$ in three dimensions for $N=1000$. We extract the coefficient of the resulting power law, $D_2~({\gamma }_P-\gamma ){}^{-\frac{3}{2}}$, and average over realizations to obtain Fig. 5. The left panel is in linear scale while the right panel is in logarithmic scale.

Figure 5

The mean of the product of the contractions, ${\mathrm{â}}_P^3{\mathrm{b̂}}_P$, as a function of system size $N$. For the calculation, the weak observed dependence is taken as no dependence: ${\mathrm{â}}_P^2~N^0$.

Figure 6

Distributions of $D_1$ measured for each instance in our ensemble of quenched amorphous solids. As expected, the width of these distributions decays with increasing system size.

Figure 7

Distributions of $D_2$ measured for each instance in our ensemble of quenched amorphous solids. As expected, the width of these distributions does not decay with increasing system size.

Figure 8

Distributions of $D_3$ measured for each instance in our ensemble of quenched amorphous solids.

Figure 9

Distributions of $D_3$ rescaled by the estimated $N$ dependence; cf. Eq. (51). The data collapse is in excellent agreement with the predicted scaling behavior. We find the best collapse with ${\zeta }_{\text{2D}}\approx 0.62$ for the two-dimensional data and ${\zeta }_{\text{3D}}\approx 0.56$ for the three-dimensional data.

Figure 10

Distributions of $X_2/N^{\frac{3}{1+\theta }}$, calculated using $\theta =2.2$. See text for details of statistical model.

Figure 11

Distributions of $X_4/N^{\frac{7}{1+\theta }}$, calculated using $\theta =2.2$. See text for details of statistical model.