Breakdown of Heterogeneous Materials

We discuss the threshold activated extremal dynamics that is prevalent in the breakdown processes in heterogeneous materials. We model such systems by an elastic spring network with random breaking thresholds assigned to the springs. Results are obtained from molecular dynamics simulation of the system under constant stress and constant strain conditions. We find that the distribution $P(m)$ of the avalanches of size $m$, caused by the rupturing of the springs till the failure of the network, decays as a power-law: $P(m) \sim m^{-\alpha}$, where $\alpha$ can be closely approximated to 5/2. The average avalanche size $<m>$ diverges as $<m> \sim (F_c - F)^{-1/2}$ close to the stress $F_c$ at which the total failure of the network occurs. We study the time evolution of the breakdown process: we find that the bonds rupture randomly over the network at initial times but the rupturing becomes highly correlated at late times to give rise to a well-defined macroscopic crack.