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

Abstract

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 interlamina 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 logarithmic͒ recovery ͑increase͒ 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-point͒ memory.

Nonlinear elasticity in rocks: A comprehensive three-dimensional description

“…Depending on the strain amplitude, there exist different regimes in which the variation of the resonance frequency with strain is dominated by classical nonlinearity, slow dynamics, or both [21,22]. These observations have been described theoretically through a number of models derived from 1D theory of elasticity [13,15,[23][24][25]. However, as soon as the geometry of the sample or testing conditions deviate from the 1D assumption, these models are no longer applicable [6,26].…”

“…Additionally, hysteresis and slow dynamics are treated as two independent mechanisms whereas it has been shown that they are strongly related [37]. A number of models have been proposed where such connection between hysteresis and slow dynamics is accounted for [24,25,38,39]. However, these models are also all derived from 1D theory of elasticity.…”

Nonlinear elastodynamics in micro-inhomogeneous solids observed by head-wave based dynamic acoustoelastic testing

“…Acoustically induced reversible conditioning was suggested to originate from local heating of internal micro-contacts due to thermoelasticity 17 or from a temporary and fully reversible increase in the concentration of ruptured cohesive bounds as described by the "soft-ratchet" model. 18 …”

Dynamic acoustoelastic testing of weakly pre-loaded unconsolidated water-saturated glass beads

“…This may be another consequence of relaxation phenomena. Alternatively, the so-called "softratchet" concept introduced by Vakhnenko et al 33 to model the nonlinear elastic behavior of consolidated granular rocks may explain both dynamic asymmetry and relaxation after the end of the LF perturbation (also called slow-dynamics) observed in water-saturated unconsolidated glass beads. Different kinetics in the opening and closing of cracks or intergrain contacts provide a physical mechanism that breaks the symmetry under sinusoidal dynamic loading.…”