An Improved Thermodynamic Energy Equation for Stress–Dilatancy Behavior in Granular Soils
Abstract
1. Introduction
2. Energy Equation and First Law of Thermodynamics
3. Energy Equation and Stress–Dilatancy Relationship
3.1. Consider Only Friction Dissipation in Energy Equation
3.2. Consider Plastic Free Energy in Energy Equation
3.3. Consider Crushing in Energy Equation
4. Proposed Energy Equation
5. Discussion
5.1. Up-Hook and Down-Hook
5.2. Model Parameters
- (1)
- Frictional parameter (Figure 10a): Crushable sands (e.g., calcareous or coral) exhibit higher values of the frictional parameter compared to uncrushed sands (e.g., quartz or silica). This increase is primarily attributed to the angular particle shapes generated during crushing, which significantly enhance interparticle friction and resistance to sharing.
- (2)
- Dilation parameter (Figure 10b): The dilation parameter does not appear to be significantly influenced by particle crushing. Instead, its value is primarily governed by the initial density state of the soil. For dense sands, typically ranges from 0 to 0.6, reflecting a greater potential for dilation during shearing due to a tightly packed structure and stronger force chains. In contrast, for loose sands, generally ranges from 0 to 0.3, as the loosely arranged particles limit dilative behavior.
- (3)
- Decay of functions of and (Figure 10c): At , the magnitude of and =. The magnitude of is large than , and it also decays fast at . In Figure 10c, the red curve (marked 1) represents , which reflects energy release due to the loss of interlocking. Since interlocking deteriorates quickly during shearing, starts large and decays sharply.The blue curve (marked 2) corresponds to energy release from force-chain buckling and particle rearrangement (negative); this process decays much more gradually than . The blue curve (marked 3) represents energy from particle crushing. Because the rate of particle crushing is most significant at the early stage of shearing, the rate decays sooner compared to the blue curve (marked 2) and becomes negligible after the peak of stress.
- (4)
- Parameter (Figure 10d): The value of is typically positive for crushed sands and negative for uncrushed sands. Crushed sands tend to exhibit larger absolute values, indicating a more substantial deviation between the phase transition state and the critical state. This reflects the increased resistance mobilized in crushable soils due to particle fragmentation and angularity effects.
- (5)
- Decay rate ratio ( (Figure 10e): The ratio ( quantifies how much faster the interlocking degradation term decays compared to the rate of decay of the force chain or crushing . While the ratio can reach up to 20, it is generally higher for crushed sands, indicating a sharper and more immediate decay of in crushable soils. This suggests that the interlocking in crushable sands is less stable and more sensitive to shear-induced degradation.
- (6)
- Initial dilation parameter (Figure 10f): The parameter , which represents the initial rate of dilation at zero shear strain, appears to be not significantly affected by particle crushing. Instead, it is more strongly influenced by the confining stress and the initial relative density of the sand. As shown in Figure 10f, generally increases with increasing confining stress, regardless of whether the sand is crushed or uncrushed. This trend suggests that under higher confinement, the soil structure develops greater resistance to volumetric contraction, leading to more pronounced dilation upon shearing.
5.3. Stress–Strain Curves Signatures
6. Summary and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Elastic Component of the Strain
References
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Author | Soil Type | σ3 | Test Conditions | Dr (%) | Size Range |
---|---|---|---|---|---|
Yu (2017) [55] | Silica sand and coral sand | 0.2–0.5 (MPa) | Dense state, precrushed samples | - | 0.1 to 2 mm |
Yao et al. (2024) [56] | Coral sand | 0.5–0.8 (MPa) | Dense state | 80 | 0.707 mm |
Norris et al. (1997) [39] | Nevada sand | 0.01–0.25 (MPa) | Loose and very loose state (e > emax) | ≈15/(e > emax) | 0.14 mm |
Hassanlourad et al. (2008) [57] | Calcareous sand | 0.05–1.2 (MPa) | Loose and dense state | 20/80 | - |
Yilmaz et al. (2023) [40] | Binary quartz sand with fines | 0.2 (MPa) | Dense state and different fines content | 97 | 0.096 to 1.086 mm |
a | 1.3 | 1.15 | 0.35 | 0.9 | 0.75 | −0.108 |
b | 1.32 | 1.21 | 0.25 | 0.2 | 3 | −0.213 |
c | 1.49 | 1.76 | 0.25 | 0.1 | 1.1 | −0.457 |
d | 1.54 | 1.86 | 0.45 | 0.05 | 2.5 | −0.783 |
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Chang, C.S.; Chao, J. An Improved Thermodynamic Energy Equation for Stress–Dilatancy Behavior in Granular Soils. Geotechnics 2025, 5, 43. https://doi.org/10.3390/geotechnics5030043
Chang CS, Chao J. An Improved Thermodynamic Energy Equation for Stress–Dilatancy Behavior in Granular Soils. Geotechnics. 2025; 5(3):43. https://doi.org/10.3390/geotechnics5030043
Chicago/Turabian StyleChang, Ching S., and Jason Chao. 2025. "An Improved Thermodynamic Energy Equation for Stress–Dilatancy Behavior in Granular Soils" Geotechnics 5, no. 3: 43. https://doi.org/10.3390/geotechnics5030043
APA StyleChang, C. S., & Chao, J. (2025). An Improved Thermodynamic Energy Equation for Stress–Dilatancy Behavior in Granular Soils. Geotechnics, 5(3), 43. https://doi.org/10.3390/geotechnics5030043