Physical models of seismic attenuation measurements in the lab

Abstract

Classical continuum mechanics with dissipation allows the description of observed creep and phase-lag attenuation effects in solids. The frequency-dependent Q or time dependent moduli, compliances, or creep functions which are often used to describe such observations may be empirical characteristics reflecting not only the properties of the materials but also the dimensions and shapes of the samples.

The theoretical paradigm employed in this study is strongly different from the conventional, Q-based (often called “viscoelastic”) model. Instead of a single, but arbitrarily frequency-dependent Q attributed to a solid, a number of specific physical parameters of energy-dissipation mechanisms (such as viscosity or thermoelasticity) are considered. The model is based on first physical principles and focuses on inverting for the intrinsic (time- and frequency-independent) properties of the material.

The observed frequency-dependent Q’s or time-dependent creep (“memory”) functions are generally explained by the non-linearity of solid viscosity, which can be described by selecting the Lagrangian dissipation function. This fundamental conclusion was suggested as long ago as by Knopoff (1964) but appeared to be little developed since. I only consider a specific, power-law form of this function, and show that it is consistent with the strain-rate dependence of effective viscosity used in geodynamics. Power-law nonlinearity of solid viscosity combined with thermoelastic effects allows quantitatively predicting all key observations, such as creep, stress-strain phase lags in torsional and longitudinal oscillations, and broadening of spectral amplitude peaks near resonance. Analytical and numerical modeling of longitudinal-oscillation phase-lag measurements in Plexiglas cylinders suggest the value of rheological exponent approximately 0.56. This is interpreted as a “near-dry” internal friction in solids. The physical models of internal friction also suggest methods for inverting for the in situ dissipation properties of materials. Finally, the new models suggest several ways for enhancing the theoretical knowledge about the physical properties of Earth materials.