Irreversible thermodynamic and viscoelastic model for power-law relaxation and attenuation of rocks

Abstract

A constitutive law for viscoelastic behaviour of rocks is derived from irreversible thermodynamics. To this model, two specific parameters are introduced; one is an internal state variable which is a variable concerning the microstructures such as defects in crystals or microcracks, and the other is a temperature reduced time obtained by normalizing the various temperature behaviours. A large number of internal state variables have the respective relaxation times and show the respective time evolutions, while a set of the time evolutions generates temporal power-law behaviour of rocks. The time evolutions of internal states are regarded as dynamics of elements of the generalized Maxwell model, and the stress–strain relation is represented by a response function following a temporal power-law in terms of linear system theory. This relation is inversely formulated to investigate the source field from output data. This model enables us to explain experimentally-based constitutive laws for transient and steady-state behaviour of rocks (e.g., lherzolite) following a temporal power-law and for attenuation behaviour of polycrystals (e.g., olivine) represented by a relation between the quality factor and frequency. Both laws show power-laws on deformation time or frequency depending on the fractal structure in polycrystals or rocks, and the experimental high-temperature behaviours can be extrapolated to long deformation time or high frequency behaviour.