Research on unified analytical solution and corresponding time-effect deformation characteristics of rock mass viscoelastic deformation under rigid and flexible bearing plate tests

Describing the phenomena and laws of rock mass-aging deformation is an important research area on bearing plate tests. Presently, no unified analytical solution exists for characterizing time-dependent displacements of flexible and rigid circular bearing plate tests. Because of this deficiency, based on the rigid and flexible bearing plate rock mass rheology tests and according to the viscoelastic theory, the unified time-dependent deformation analysis solution of the half-space viscoelastic body under a normal uniform load on the surface is derived. Based on this derivation, the viscoelastic analytical solution of rock mass-aging deformation and the elastic aftereffect displacement equation for the Maxwell, Kelvin–Voigt, and Burgers models, respectively, were determined. The viscoelastic characteristics of rock mass-aging deformation under bearing plate tests and the influence law of viscoelastic parameters on rock mass-aging deformation are discussed. The results show that the viscoelastic analytical solution of the rock mass-aging deformation at the center of the rigid circular bearing plate is π/4 that of the flexible circular bearing plate test. Also, the viscoelastic analytical solution of the time-dependent deformation of the rock mass at the edge of the flexible circular bearing plate is 2/π that at the center of the bearing plate. However, the evolution trend and law of the time-dependent deformation of the rock mass at the center of both circular bearing plate tests are consistent. Furthermore, the viscoelastic analytical solutions of the rock mass deformation at the center of the circular bearing plate obtained by the above three rheological models have elastic aftereffect characteristics. Precisely, although the Maxwell rheological model does not have elastic aftereffect characteristics, the viscoelastic analytical solution equation for rock mass deformation at the center of the circular bearing plate test obtained from this model has an exponential term related to time, which results in the gradual decay of the rock mass deformation with time after unloading, i.e., showing elastic aftereffect characteristics. When the Maxwell and the Burgers models are used, the rock mass under the bearing plate test condition has the same final residual deformation value $\omega \frac{q{R}_{0}}{2}\frac{{t}_{1}}{{\eta }_{0}}$ after unloading. However, when the Kelvin–Voigt model is used, no residual deformation of the rock mass exists after unloading under the bearing plate test conditions, which is consistent with the law of the elastic aftereffect phenomenon of the Kelvin–Voigt model.