Correlation between avalanches and emitted energies during fracture with variable stress release range

We observe the failure process of a fiber bundle model with a variable stress release range, $\gamma$, higher the value of $\gamma$ lower the stress release range. By tuning $\gamma$ from low to high, it is possible to go from the mean-field (MF) limit of the model to local load sharing (LLS) where local stress concentration plays a crucial role. In the MF limit, the avalanche size $s$ and energy $E$ emitted during the avalanche are highly correlated producing the same distribution for both $P(s)$ and $Q(E)$: a scale-free distribution with a universal exponent -5/2. With increasing $\gamma$, the model enters the LLS limit. In this limit, due to the presence of local stress concentration such correlation $C(\gamma)$ between $s$ and $E$ decreases where the nature of the decreases depends highly on the dimension of the bundle. In 1d, the $C(\gamma)$ stars from a high value for low $\gamma$ and decreases towards zero when $\gamma$ is increased. As a result, $Q(E)$ and $P(s)$ are similar at low $\gamma$, an exponential one, and then $Q(E)$ becomes power-law for high-stress release range though $P(s)$ remains exponential. On the other hand, in 2d, the $C(\gamma)$ decreases slightly with $\gamma$ but remains at a high value. Due to such a high correlation, the distribution of both $s$ and $E$ is exponential in the LLS limit independent of how large $\gamma$ is.