Hysteresis and two‐dimensional nonlinear wave propagation in Berea Sandstone

Abstract

We have further developed a two‐dimensional endochronic constitutive model applicable to rock in the intermediate strain regime, i.e., approximately 10 −6 to 10 −3 , where nonlinear losses, pulse distortion, and harmonic distortions have been documented in the laboratory. An accurate and efficient computation algorithm has been established and used to fit a set of triaxial laboratory data on Berea Sandstone, obtained by G. N. Boitnott of New England Research, Inc., including nonlinear hysteresis for both hydrostatic compression and shear deformation. The constitutive model has been incorporated into a two‐dimensional pseudospectral scheme for simulating nonlinear wave propagation. We compare our numerical pseudospectral scheme with the perturbation solutions to nonlinearly elastic plane wave propagation and find that the two solutions are in close agreement. Applying the numerical scheme to the endochronic constitutive model for Berea Sandstone, we find the following results, which confirm and extend previous results obtained with a one‐dimensional model: (1) nonlinear propagation from a monochromatic source excites odd harmonics of the source frequency, (2) the interaction of two monochromatic wave fields leads to energy transfer to other bands corresponding to various combinations of the two source frequencies, (3) amplitudes of the harmonics change with distance and show a trade‐off between nonlinearity, intrinsic attenuation (hysteresis), and geometrical spreading, (4) for a broadband pressure source, energy is transferred from the principal band to a higher band at the expense of the components at intermediate frequencies; this is diagnostic of nonlinear interactions, and (5) the harmonic amplitude and energy transfer increase nonlinearly with strain amplitude; i.e., higher strain levels lead to higher energy exchange. These simulations of harmonic generation, interaction, and band expansion, based on a model developed from quasistatic experimental observations, agree with dynamic experimental observations.