Analytical Relation between b-Value and Electromagnetic Signals in Pre-Macroscopic Failure of Rocks: Insights into the Microdynamics’ Physics Prior to Earthquakes

Field measurements in subduction regions have revealed the presence of non-seismic pre-earthquake signals such as electromagnetic or acoustic emission, gas liberation, changes in Earth’s surface temperature, changes at the ionospheric level, or fluid migration. These signals are commonly associated with impending earthquakes, even though they often rely solely on temporal and spatial correlations in impending earthquake zones without a comprehensive understanding of the underlying lithospheric processes. For example, one criticism is the measurement of increasing electromagnetic signals even in the absence of observable macroscopic stress changes, which challenges the conventional understanding that macroscopic stress changes are the primary energy source for non-seismic pre-earthquake signals. To address this gap, rock experiments provide valuable insights. Recent experiments have shown that rocks can become electrified under constant macroscopic stress changes, accompanied by a decrease in the b-value, indicating multiscale cracking. This suggests the existence of small-scale dynamics that generate electromagnetic signals independently of large-scale stress variations. In that sense, multiscale thermodynamics offers a valuable perspective in describing this multiscale phenomenon. That is why the main goal of this work is to demonstrate that the electromagnetic signals before macroscopic failures are not independent of the cracking generation because the origin of both phenomena is the same. In particular, we present analytical equations that explain the physical connection between multiscale cracking, the generation of electromagnetic signals, and its negative correlation with acoustic emission before the macroscopic failure of rocks even when the macroscopic load is constant. In addition, we also show that the thermodynamic fractal dimension, which corresponds to the global parameter that controls the cracking process, is proportional to the b-value when the large-scale crack generation is considerably larger than the small-scale cracks. Thus, the decreases in the b-value and the increases in the electromagnetic signals indicate that rocks irreversibly prepare to release energy macroscopically. These findings could be related to the dynamics at lithospheric scales before earthquakes.