On Critical States, Rupture States and Interlocking Strength of Granular Materials
Abstract
:1. Introduction
2. Results
2.1. Critical-Compaction Surface
2.2. Critical-State Surface
2.3. Rupture Surface
− [m − a(ω)·g(ω)·χ]cos ψ·α0’ − g(ω)·χ·cos ψ·α0’,
for pf’ ≤ p’ ≤ pcs’.
3. Discussion
3.1. Strength
3.2. Dilation
- Define interlocking in terms of the difference between the critical mean-normal effective stress and the mean-normal effective stress in place of Taylor’s ratio.
- Distinguish any virtual cohesion from the contributions due to over-compaction.
3.3. Comparison
4. Materials and Methods
5. Conclusions
Acknowledgments
Conflicts of Interest
Glossary/Nomenclature/Abbreviation
Parameters | |
α’ | normal compaction strength, maximum mean-normal effective stress |
α0’ | minimum mean-normal effective stress for well-defined flexibilities |
ν | specific volume |
νcs | specific volume at critical compaction |
σ1’ | major principal effective stress |
σ2’ | intermediate principal effective stress |
σ3’ | minor principal effective stress |
σ’ | average of the major and minor principal effective stresses |
σf’ | σ’ at the intersection of the fracture and rupture surfaces |
σnr’ | normal effective stress component on the rupture plane |
σns’ | normal effective stress component on the slip plane |
σcs’ | average of major and minor principal effective stresses at critical state |
τr | shear strength component along the rupture plane |
τs | shear strength component along the slip plane |
ω | shearing mode |
p’ | mean-normal effective stress |
p0’ | reference mean-normal effective stress |
pf’ | p’ at the intersection of the fracture and rupture surfaces |
pcs’ | critical-compaction strength—p’ at critical state |
r | radius of Mohr’s circle of stress |
Material Properties | |
β | index of ln-ln normal compaction line in ν-p’ space |
Γ(ω) | specific volume at p0’ on the continuous-flow line for ω |
Γ | specific volume at p0’ on the continuous-flow line—a material constant |
ξ | index of ln-ln swelling line in ν-p’ space |
κ | index of semi-ln swelling line in ν-p’ space |
λ | index of semi-ln normal compaction line in ν-p’ space |
φ | angle of internal friction on the slip plane |
ψ | angle of internal friction on the rupture plane |
χ | coefficient of interlocking strength |
a(ω) | phase-transition function |
cr | virtual cohesion along the rupture plane |
cs | virtual cohesion along the slip plane |
f(p’,ν) | apparent flexibility—slope of the normal compaction line |
fc(p’,ν) | centric flexibility—slope of the swelling line |
g(ω) | intermediate principal effective stress factor |
m | coefficient of compaction influence |
M | coefficient of internal friction on the slip plane—Cambridge research program |
N | specific volume at p0’ on the normal compaction line |
P | coefficient of interlocking—Cambridge research program |
R | coefficient of internal friction on the rupture plane—Cambridge research program |
Χ | coefficient of interlocking strength—Cambridge research program |
Terzaghi’s Implementation—equivalent terms are shown in parentheses | |
α | normal compaction strength |
α0 | coherence threshold for well-defined flexibilities |
σ | normal stress under normally compacted conditions (σns) |
σ’ | normal stress under over-compacted conditions (σnr) |
σ1 | major principal stress |
σ3 | minor principal stress |
σcs | average of major and minor principal stresses at critical state |
τ | shear strength under normally compacted conditions (τs) |
τ’ | shear strength under over-compacted conditions (τr) |
φ | angle of internal friction under normally compacted conditions (φ) |
φ’ | angle of internal friction under over-compacted conditions (‘true’) (ψ) |
φf | angle of internal friction in Terzaghi’s Equation (5.3) [5] |
c | cohesion under normally compacted conditions (cs) |
c’ | effective cohesion under over-compacted conditions (‘true’) (cr + m·(α − α0)) |
c0’ | virtual cohesion under over-compacted conditions (cr) |
N | gradient of the effective cohesion relation in Terzaghi’s Equation (5.3) [5] |
s | shear strength in Terzaghi’s implementation |
Appendix A. Terzaghi’s Theory of Strength
= c0’·cos φ’ + [m·cos φ’ − a(ω)·g(ω)·(sin φ − sin φ’)]·α
− [m·cos φ’ − a(ω)·g(ω)·(sin φ − sin φ’)]·α0
− g(ω)·(sin φ − sin φ’)·α0.
+ g(ω)·(sin φ − sin φ’)·α0.
+ a(ω)·g(ω)·(sin φ − sin φ’)·(α − α0)·cos φ’.
cos φ·cos φ’.
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Feature | Present Theory | Jenike–Shield | Schofield | Collins et al. |
---|---|---|---|---|
Non-teleological | Yes | Yes | No | No |
Reference stress | α’ | α’ = pcs’ = σcs’ | pcs’ | pcs’ |
Shearing mode | Yes | Yes | No | No |
Cohesion | Yes | Yes | No | No |
Range of σ2’ | σ 3’ ≤ σ2’ ≤ σ1’ | σ2’ = (σ1’ + σ3’)/2 | σ2’ = σ3’ | σ2’ = σ3’ |
Locus of critical states is a | Surface | Surface | Line | Line |
Strain rates at critical state | Uncoupled | Uncoupled | Coupled | Coupled |
Allows rate dependence | Yes | Yes | No | No |
Elasto-plastic, rigid-plastic | Elasto-plastic | Rigid-plastic | Elasto-plastic | Elasto-plastic |
Parallel flow | Yes | Yes | Yes | Yes |
Non-parallel flow | Yes | Yes | No | No |
New state parameter | α’ | α’ | ? | ξ |
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Szalwinski, C.M. On Critical States, Rupture States and Interlocking Strength of Granular Materials. Materials 2017, 10, 865. https://doi.org/10.3390/ma10080865
Szalwinski CM. On Critical States, Rupture States and Interlocking Strength of Granular Materials. Materials. 2017; 10(8):865. https://doi.org/10.3390/ma10080865
Chicago/Turabian StyleSzalwinski, Chris M. 2017. "On Critical States, Rupture States and Interlocking Strength of Granular Materials" Materials 10, no. 8: 865. https://doi.org/10.3390/ma10080865