Hysteresis and two-dimensional nonlinear wave propagation in Berea Sandstone

We have further developed a two-dimensional endochronic constitutive model applicable to rock in the intermediate strain regime, i.e., approximately 1 0(-6) to 10(-3), where nonlinear losses, pulse distortion, and harmonic dis tortions have been documented in the laboratory. An accurate and efficient computation algorithm has been established and used to fit a set of triaxia l laboratory data on Berea Sandstone, obtained by G. N. Boitnott of New Eng land Research, Inc., including nonlinear hysteresis for both hydrostatic co mpression and shear deformation. The constitutive model has been incorporat ed into a two-dimensional pseudospectral scheme for simulating nonlinear wa ve propagation. We compare our numerical pseudospectral scheme with the per turbation solutions to nonlinearly elastic plane wave propagation and find that the two solutions are in close agreement. Applying the numerical schem e to the endochronic constitutive model for Berea Sandstone, we find the fo llowing results, which confirm and extend previous results obtained with a one-dimensional model: (1) nonlinear propagation from a monochromatic sourc e excites odd harmonics of the source frequency, (2) the interaction of two monochromatic wave fields leads to energy transfer to other bands correspo nding to various combinations of the two source frequencies, (3) amplitudes of the harmonics change with distance and show a trade-off between nonline arity, intrinsic attenuation (hysteresis), and geometrical spreading, (4) f or a broadband pressure source, energy is transferred from the principal ba nd to a higher band at the expense of the components at intermediate freque ncies; this is diagnostic of nonlinear interactions, and (5) the harmonic a mplitude 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 d eveloped from quasistatic experimental observations, agree with dynamic exp erimental observations.