Soft-ratchet modeling of end-point memory in the nonlinear resonant response of sedimentary rocks

We developed and thoroughly examined a model of longitudinal vibrational resonance in bar-shaped sedimentary rocks; these materials exhibit memory that originates from an essential asymmetry in processes of rupture and recovery of intergrain and interiamina cohesive bonds. The theory relies on an appropriate isolation and an adequate formalization of two mutually dependent subsystems, namely, a subsystem of ruptured bonds and a subsystem of internal longitudinal displacements. The subsystem of ruptured bonds is shown to be of a soft-ratchet type, so that its response to an alternating internal stress is characterized by broken symmetry and appears as nonzero long-term temporal and spatial changes in the concentration of ruptured bonds. The internal stress is generated by an alternating external drive acting both directly through the subsystem of longitudinal displacements and indirectly through temporal and spatial modifications of Young's modulus due to changes in concentration of ruptured bonds. The scheme reproduces the main experimental effects by using the simplest linear form of attenuation in an elastic subsystem and realistic assumptions about the stress-strain relation. In particular, it correctly describes: hysteretic behavior of a resonance curve on both its upward and downward slopes; linear softening of resonant frequency with increase of driving level; gradual (almost logarithmic) recovery (increase) of resonant frequency at low dynamical strains after the sample was conditioned by high strains; and temporal relaxation of response acceleration amplitude at fixed frequency. These are the most interesting observations typical of forced longitudinal oscillations of sandstone bars in the nonlinear regime. Further, we are able to trace how water saturation enhances the hysteresis and simultaneously decreases the quality factor because of an increase in equilibrium concentration of ruptured cohesive bonds. We also predict theoretically a dynamical effect analogous to the widely known quasistatic effect of hysteresis with discrete (end-point) memory.