Stress relaxation constitutive model of rock based on Hausdorff derivative

Abstract

Stress relaxation is a key time-dependent response that governs the long-term stability of deep underground rock engineering. In this study, staged uniaxial and triaxial stress relaxation tests on sandstone were carried out using an MTS815.02 triaxial rheological servo system. Based on the Hausdorff derivative, a fractal-order dashpot was introduced to replace the Newtonian dashpot in the classical Poynting–Thomson model, yielding a nonlinear viscoelastic relaxation model with a time-fractal parameter α. By decomposing stress/strain into spherical and deviatoric parts, the model was further extended to a three-dimensional stress state and verified against triaxial relaxation data. Model parameters were identified by the Levenberg–Marquardt algorithm, and the fitting correlation coefficient for all strain levels exceeded 0.95. A parameter sensitivity analysis quantified the roles of elastic moduli, viscosity, and α in controlling the initial stress level, relaxation magnitude, and relaxation rate. The proposed formulation provides a practical constitutive description for relaxation-dominated deformation of sandstone under high in-situ stress and offers a basis for time-dependent stability assessment and support design in deep rock engineering.