Unified critical state model for quartz sand and calcareous sand under different particle gradations

Quartz sand and calcareous sand are extensively utilized as fundamental construction materials in inland regions and South China Sea island reefs, respectively. These two types of sand exhibit markedly different particle gradation characteristics, which play a pivotal role in determining their mechanical behavior, particularly the evolution of the critical state line (CSL). Establishing a unified critical state mathematical model that accurately captures the influence of particle gradation is of paramount importance for addressing practical engineering challenges such as foundation design, slope stability analysis, and infrastructure development in diverse geological environments. This study aims to bridge the gap between the behavior of quartz sand and calcareous sand by developing a comprehensive framework that incorporates particle gradation effects. The experimental investigation employed a systematic approach using triaxial isotropically consolidated drained shear tests on carefully prepared specimens of Fujian quartz sand and calcareous sand collected from a reclaimed island reef in the South China Sea. The particle gradation parameter (I_G) was introduced as a robust quantitative indicator to characterize particle size distributions. The experimental program was designed to encompass three distinct gradation conditions (I_G = 0.603, 0.686, and 0.773) and four relative densities (D_r = 30%, 50%, 70%, and 90%) to ensure comprehensive coverage of possible field conditions. The comprehensive results from the experimental program revealed several significant findings. As I_G increased from 0.603 to 0.773, both quartz and calcareous sands demonstrated substantial improvements in peak strength characteristics. Specifically, the peak strength enhancement was more pronounced in calcareous sand than in quartz sand under identical conditions. Concurrently, the axial strain required to reach peak strength decreased significantly, indicating a transition towards more brittle behavior. The critical state lines of both sand types exhibited downward shifts with increasing I_G values; however, while quartz sand maintained essentially constant critical state line slopes due to minimal particle breakage, calcareous sand showed remarkable changes in both the position and orientation of its critical state line. Calcareous sand exhibited substantial particle breakage under increasing stress levels, with the relative breakage (B_r) showing a strong correlation with both consolidation pressure and initial gradation. The differential relative breakage between different I_G values increased with rising stress levels, while the corresponding differential critical void ratios decreased progressively. This interdependent relationship directly contributed to the observed rotation of the critical state line, with the slope decreasing as I_G increased from 0.603 to 0.773. Based on these experimental observations, a unified quantitative relationship integrating the particle gradation parameter, relative breakage, and critical void ratio was developed through rigorous mathematical formulation. The model parameters were calibrated using advanced optimization techniques, resulting in excellent agreement between calculated and measured critical void ratios with ratios ranging from 0.85 to 1.15. The constitutive model, built upon the state-dependent dilatancy framework, successfully incorporates the effects of particle gradation through carefully derived mathematical expressions that account for the complex interactions between particle breakage, density conditions, and stress levels. Validation using independent experimental data from the literature confirmed the model's robustness and predictive capabilities across different loading conditions and initial states. The model demonstrates particular strength in capturing the transition from contractive to dilative behavior and accurately predicting the stress-strain response under various combinations of particle gradation, density, and confinement conditions. The research findings have substantial implications for geotechnical engineering practice, particularly for projects involving sand foundations, embankments, and offshore structures where particle gradation variations are inevitable. The developed model provides engineers with a practical tool for more accurate prediction of soil behavior, leading to improved design efficiency and enhanced safety margins. Future research directions include extending the model to incorporate time-dependent effects, cyclic loading conditions, and partial saturation scenarios to address an even wider range of practical applications.