Packing Densification Response-Constrained Fractal Characterization and Compaction Performance Evaluation of Widely...

Not all particle-size fractions in widely graded granular materials contribute equally to compaction densification. For non-ideal particle-size distributions (PSDs) with local deviations or fine-end disturbances, the full-range fractal index may be influenced by particle-size fractions that contribute weakly to densification and, therefore, may not consistently represent the maximum dry density response. To address this problem, this study proposes a response-constrained truncation framework to identify a more effective PSD fitting range for fractal characterization. First, 20 concave and S-shaped PSDs from previous experiments were re-analyzed to compare full-range and truncated indices. Then, 21 progressively truncated specimens derived from three standard fractal PSDs were tested by relative density experiments. A unit-mass densification contribution coefficient, η_j, was defined from adjacent maximum dry density differences and particle-fraction mass contents. The η_j-d responses exhibited unimodal patterns, and the transition diameter d_c shifted with PSD coarseness. For the two material sources, replacing the full-range index with the truncated index increased the R² values between the fractal index and maximum dry density from 0.195 to 0.886 and from 0.191 to 0.856, respectively. A continuous percentile search showed that the optimal characteristic scale was concentrated near q ≈ 30, with a robust common optimum of q = 30.53. Sensitivity analysis for β = 0.85-0.95 indicated that 0.225d₃₀ falls within the transition region from highly effective filling to reduced densification efficiency. Accordingly, d_L = 0.225d₃₀ is proposed as a preliminary engineering estimate of the lower fitting limit for non-ideal PSDs. The framework is intended for widely graded materials whose full-range fractal parameters are inconsistent with compaction response.