Three‐Dimensional Modeling of Hysteresis in Rocks

Hysteresis in rock is considered to be as one of the basic possible material behaviors. McCall and Guyer treat rocks as materials consisting of hysteretic mesoscopic elastic units (HMEU). Hysteresis with discrete memory can be well reflected by the density of HMEU in Preisach‐Mayergoyz space (PM space). However, this model is limited to one‐dimensional case and unable to obtain all the six components of strain tensor. We revisit and extend their model to three‐dimensional case by introducing a type of initially closed crack, and by redefining the hysteretic behavior of the initially open/closed mesoscopic cracks. Together with the directional probability density function E(n), our analysis leads to an explicit expression of the strain tensor of a rock under triaxial loading/unloading. The ratio of closed cracks, the direction of cracks and a parameter related to the equilibrium length of cracks are included in this three‐dimensional model. Plain Language Summary Hysteresis is a natural property of rock. This property can be well reflected by treating each mesoscopic crack in rock as a hysteretic unit. All those mesoscopic cracks can be presented in Preisach‐Mayergoyz space (PM space). The hysteretic strain can be obtained for a given loading/unloading path by this way. However, the current hysteretic models are limited to one‐dimensional case and unable to get all the six components of the strain tensor. We redefine the hysteretic behavior of those mesoscopic cracks and introduce the directional probability density function E(n) to the one‐dimensional model. Finally, we give an explicit expression of strain tensor of a rock under triaxial loading/unloading. All the six components of the hysteretic strain tensor are available. The results of this paper can be regarded as a reference for the study of hysteresis in granular material. Key Points The limitations of one‐dimensional hysteretic model are discussed The hysteretic behaviors hysteretic mesoscopic elastic units are redefined A three‐dimensional model of hysteresis in rock is developed