A Gompertz-Based Statistical Damage Model for Water–Rock Interaction under Uniaxial Compression and Its Verification

Water erosion significantly degrades the mechanical strength of rock, leading to a decline in rock stability. Understanding the damage evolution characteristics and constitutive behavior of rock under water erosion is therefore crucial. This study leverages continuous damage mechanics and statistical damage theory to establish a theoretical damage evolution equation for rock subjected to coupled hydromechanical effects. The equation incorporates a Gompertz distribution function and a water action damage variable. To account for the nonlinear compaction behavior during water–rock interaction, a piecewise constitutive model is proposed, incorporating a compaction coefficient (C). The model’s validity is verified using uniaxial compression test results of coal samples subjected to varying immersion times. The findings indicate the following: (1) the theoretical damage evolution model, based on the Gompertz distribution function, exhibits an asymmetric S-shaped trend, accurately capturing the initial rapid followed by gradual deceleration of damage development; (2) water–rock interaction significantly weakens the mechanical properties of coal samples. Increasing the water–rock interaction time correlates with a downward trend in both the maximum damage evolution rate and the inflection point strain value. (3) The proposed piecewise constitutive model effectively simulates the entire uniaxial compressive stress–strain behavior of coal samples under water–rock interaction. Strong agreement between theoretical model curves and experimental data demonstrates the model’s robust simulation capability and applicability. (4) Within the constitutive model, parameters m′ and F0′ control the concavity and compaction stage length, respectively. Parameters m and ε0 influence the postpeak stress–strain curve’s drop rate and deposition. Collectively, these four parameters determine the overall shape of the constitutive model curve.