Hysteresis in elastic behavior: The connection between low-frequency response and acoustic properties of rocks

The strain response of rock to quasistatic stress cycles (e.g., 10−3 Hz) is highly nonlinear, hysteretic, and displays discrete memory. Rocks also display unusual nonlinear behavior in acoustic wave experiments (e.g., 104 Hz). Nonlinearity and hysteresis are prominent features in elastic measurements on rocks. This observation is the key to making the connection between low-frequency (quasistatic) and high-frequency (acoustic) measurements, e.g., between static modulus and dynamic modulus. A new paradigm has been developed for the description of the elastic behavior of rocks and other consolidated materials. This paradigm uses the statistical properties of an ensemble of micron-scale hysteretic mechanical units to describe the elastic response of a macroscopic piece of material. It provides a recipe for inverting stress-strain data (low-frequency data) for the distribution of hysteretic mechanical units. From this distribution, the high-frequency acoustic response of the macroscopic piece of material can be predicted. The new paradigm will be described in principle and in application. Quasistatic stress-strain data on sandstone lead to predictions for dynamic modulus and resonant response that agree well with experiment.