Nonlinear viscoelastic constitutive relation for rocks under micro-strain dynamic loading

Rocks exhibit pronounced nonlinear viscoelastic behavior such as modulus softening and loading–unloading hysteresis, even under micro-strain dynamic loading. However, classical nonlinear elastic constitutive models based on higher-order expansions of Hooke’s law show limitations in capturing these nonlinear viscoelastic features. To address this, we extend the classical nonlinear elastic framework by incorporating strain-rate terms and formulate a nonlinear viscoelastic constitutive relation in which nonlinear elastic and viscoelastic parameters jointly describe modulus softening, dynamic response, and hysteresis loops. We use a copropagating acousto-elastic testing system to acquire time series of elastic modulus variation ΔM/E and the corresponding hysteresis loops for Crab Orchard sandstone at five dynamic strain levels. We then invert these data to estimate model parameters and test the constitutive relation. The model reproduces the main nonlinear viscoelastic features observed at all strain levels and, compared with two classical nonlinear elastic models without viscoelastic terms, better captures the phase information in the ΔM/E time series and the geometry of the hysteresis loops. The proposed nonlinear viscoelastic constitutive relation provides a practical way to constrain micro-strain nonlinear viscoelastic parameters of rocks in the laboratory and offers a basis for linking laboratory measurements with field relative velocity changes monitoring to study stiffness evolution during processes such as hydrological loading and fault slow slip.