Localization of deformation and its effects on power-law singularity preceding catastrophic rupture in rocks

Three distinct length scales are involved in the deformation evolution and catastrophic rupture of heterogeneous rocks in general: two essential ones are the specimen size macroscopically and the grain size at micro-scale respectively, the other is the emerging localized band of deformation and damage. The band initiates almost nearby the peak load, and the rupture eventually occurs afterwards within the localized band. In this paper, we report that with the evolution of concentrated high strain and damage in the localized band, a power-law singularity emerges within the localized band preceding the eventual rupture. The localization of deformation imposes a spatial non-uniqueness on the power-law singularity, and then leads to a trans-scale characteristic of the power-law singularity. Based on this characteristic, it is demonstrated that the singularity presented by the global response of a whole specimen comes from the singularity of local response in the localized band. The localization and the power-law singularity are associated precursory events, spatially and temporally, respectively, before macroscopic rupture. In particular, based on the power-law singularity exhibited in the zonal areas near or across the rupture surface, a prediction of the occurrence time of catastrophic rupture can be made accordingly. This provides a practically helpful approach to the prediction of rupture, merely by means of monitoring the zonal areas adjacent to the localized band.