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Packing Densification Response–Constrained Fractal Characterization and Compaction Performance Evaluation of Widely Graded Granular Materials.

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Title: Packing Densification Response–Constrained Fractal Characterization and Compaction Performance Evaluation of Widely Graded Granular Materials.
Authors: Ren, Guo-Feng1 (AUTHOR), Wang, Xin-Qing2,3 (AUTHOR), Wang, Yi1,3 (AUTHOR), Hu, Qiu-Yue1,4 (AUTHOR), Pei, Xiang-Jun1,3,4 (AUTHOR), Zhang, Xiao-Chao2,3 (AUTHOR)
Source: Materials (1996-1944). Jun2026, Vol. 19 Issue 12, p2675. 18p.
Subjects: Particle size distribution, Granular materials, Fractal analysis, Soil density
Abstract: Highlights: A response-constrained PSD truncation framework is proposed for non-ideal PSDs. The unit-mass densification contribution coefficient identifies effective size ranges. A preliminary lower fitting limit is supported by geometry and response data. Truncated indices improve maximum dry density characterization for non-ideal PSDs. 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, ηj, was defined from adjacent maximum dry density differences and particle-fraction mass contents. The ηj-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 ≈ 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, 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. [ABSTRACT FROM AUTHOR]
Copyright of Materials (1996-1944) is the property of MDPI and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Packing Densification Response–Constrained Fractal Characterization and Compaction Performance Evaluation of Widely Graded Granular Materials.
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  Data: <searchLink fieldCode="JN" term="%22Materials+%281996-1944%29%22">Materials (1996-1944)</searchLink>. Jun2026, Vol. 19 Issue 12, p2675. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Particle+size+distribution%22">Particle size distribution</searchLink><br /><searchLink fieldCode="DE" term="%22Granular+materials%22">Granular materials</searchLink><br /><searchLink fieldCode="DE" term="%22Fractal+analysis%22">Fractal analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Soil+density%22">Soil density</searchLink>
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  Data: Highlights: A response-constrained PSD truncation framework is proposed for non-ideal PSDs. The unit-mass densification contribution coefficient identifies effective size ranges. A preliminary lower fitting limit is supported by geometry and response data. Truncated indices improve maximum dry density characterization for non-ideal PSDs. 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, ηj, was defined from adjacent maximum dry density differences and particle-fraction mass contents. The ηj-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 ≈ 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, 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. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Materials (1996-1944) is the property of MDPI and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.3390/ma19122675
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        Text: English
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      – SubjectFull: Particle size distribution
        Type: general
      – SubjectFull: Granular materials
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      – SubjectFull: Fractal analysis
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      – SubjectFull: Soil density
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    Titles:
      – TitleFull: Packing Densification Response–Constrained Fractal Characterization and Compaction Performance Evaluation of Widely Graded Granular Materials.
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            NameFull: Ren, Guo-Feng
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              Text: Jun2026
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              Y: 2026
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