{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T13:16:20Z","timestamp":1792588580964,"version":"4.5.0"},"reference-count":41,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T00:00:00Z","timestamp":1782086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Research and Development Program of the Tibet Autonomous Region","award":["XZ202401ZY0091"],"award-info":[{"award-number":["XZ202401ZY0091"]}]},{"name":"Tianfu Yongxing Laboratory Organized Research Project Funding","award":["2023KJGG05"],"award-info":[{"award-number":["2023KJGG05"]}]},{"name":"2026 Key Scientific Research Project of Sichuan Jiaotong Polytechnic University","award":["2026-KJ-11"],"award-info":[{"award-number":["2026-KJ-11"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>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, \u03b7j, was defined from adjacent maximum dry density differences and particle-fraction mass contents. The \u03b7j-d responses exhibited unimodal patterns, and the transition diameter dc 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 \u2248 30, with a robust common optimum of q = 30.53. Sensitivity analysis for \u03b2 = 0.85\u20130.95 indicated that 0.225d30 falls within the transition region from highly effective filling to reduced densification efficiency. Accordingly, dL = 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.<\/jats:p>","DOI":"10.3390\/ma19122675","type":"journal-article","created":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T08:19:52Z","timestamp":1782202792000},"page":"2675","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Packing Densification Response\u2013Constrained Fractal Characterization and Compaction Performance Evaluation of Widely Graded Granular Materials"],"prefix":"10.3390","volume":"19","author":[{"given":"Guo-Feng","family":"Ren","sequence":"first","affiliation":[{"name":"Sichuan Water Development Investigation, Design & Research Co., Ltd., Chengdu 610213, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xin-Qing","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Architecture and Civil Engineering, Sichuan Jiaotong Polytechnic University, Chengdu 611000, China"},{"name":"State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Wang","sequence":"additional","affiliation":[{"name":"Sichuan Water Development Investigation, Design & Research Co., Ltd., Chengdu 610213, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiu-Yue","family":"Hu","sequence":"additional","affiliation":[{"name":"Sichuan Water Development Investigation, Design & Research Co., Ltd., Chengdu 610213, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiang-Jun","family":"Pei","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China"},{"name":"Tianfu Yongxing Laboratory, Chengdu 610213, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiao-Chao","family":"Zhang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1111\/j.1151-2916.1930.tb16222.x","article-title":"The packing of particles","volume":"13","author":"Westman","year":"1930","journal-title":"J. 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