Rigidity-Controlled Crossover: From Spinodal to Critical Failure

Failure in disordered solids is accompanied by intermittent fluctuations extending over a broad range of scales. The implied scaling has been previously associated with either spinodal or critical points. We use an analytically transparent mean-field model to show that both analogies are relevant near the brittle-to-ductile transition. Our study indicates that in addition to the strength of quenched disorder, an appropriately chosen global measure of rigidity (connectivity) can be also used to tune the system to criticality. By interpreting rigidity as a timelike variable we reveal an intriguing parallel between earthquake-type critical failure and Burgers turbulence.

Figure 1

(a) Schematic representation of the system; (b) brittle response at $\mathrm{\Lambda }=0.4$; (c) ductile response at $\mathrm{\Lambda }=5$. Solid black lines: equilibrium path, thin black line: out-of-equilibrium paths; grey lines: metastable states. Parameters: $N=100$, $\lambda =1$, $\rho =4$.

Figure 2

(a) Ensemble averaged brittle-to-ductile transition line; (b) typical averaged stress-strain curves; (c) nonequilibrium avalanche distribution; (d) equilibrium avalanche distribution. Parameters: $N={10}^6$, $\lambda =1$. The avalanche distributions was averaged over ${10}^4$ realizations.

Figure 3

Finite size crossover associated with brittle to ductile transition. Each curve gives the value of the scaling exponent averaged over 250 realizations. Parameters $\lambda =1$ and $\nu =0.873$ (critical value at $\rho =4$). The exponents and the uncertainty were computed using the maximum likelihood method [65].

Figure 4

(a) Time (rigidity) evolution of the randomly distributed initial Burgers data ${\sigma }_0(ϵ)$ (red) at $\rho =2$, and $\lambda =1$; black line corresponds to ${\nu }_{*}=\exp (1/\rho +1)/\rho$. (b) Shock merging with critical complexity appearing at $\nu ={\nu }_{*}$. Thick black line shows the shock in the averaged system which emerges in the limit $N\to \infty$.

Figure 5

Time (rigidity) evolution of the normalized standard deviation for the number of shocks. The statistics was obtained from $K=1000$ realizations of the quenched disorder with $\rho =3$, and $\lambda =1$. Inset plots: (a) $\nu =0.1$ (b) $\nu =1$; (c) $\nu =6$.