Abstract
Critical state line (CSL) is the central concept in soil mechanics. A series of true triaxial compression tests under the constant-\({p}'\) and constant-b loading condition were carried out to investigate the CSL of a coarse granular soil. It was observed that the intermediate principal stress ratio (i.e., the b-value) greatly influenced the CSLs in both \(q{-}{p}'\) and \(e{-}{p}'\) spaces. The CSL slope in the \(q{-}{p}'\) space decreased with an increase in b-value. The intercept and gradient of the CSL in the \(e{-}{p}'\) space decreased with increasing b-value. CSLs incorporating the effects of the b-value in \(q{-}{p}'\) and \(e{-}{p}'\) spaces were extended to three-dimensional critical state surfaces (TCSSs) in \(q{-}{p}'{-}b\) and \(e{-}{p}'{-}b\) spaces. Two empirical equations were proposed for the two TCSSs in \(q{-}{p}'{-}b\) and \(e{-}{p}'{-}b\) spaces, respectively. The predictions by the two equations were in good agreement with the corresponding experimental data. The relationship between the excess friction angle (the difference between the peak state and critical state friction angles) and initial state parameter was influenced by the b-value. However, the relationship between the maximum dilatancy and initial state parameter was independent of the b-value.
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Abbreviations
- b :
-
Intermediate principal stress ratio
- \(d_\mathrm{M}\) :
-
Maximum particle size (unit: mm)
- \(C_\mathrm{u}\) :
-
Coefficient of uniformity
- \(C_\mathrm{c}\) :
-
Coefficient of curvature
- \(F_\mathrm{c}\) :
-
Fines content (unit: %)
- \({\sigma }'_{1} ,{\sigma }'_{2}\) and \({\sigma }'_{3}\) :
-
Major, intermediate and minor effective principal stresses, respectively (unit: kPa)
- \({p}'\) :
-
Mean effective stress (unit: kPa)
- \({p}'_{\mathrm{cs}}\) :
-
Mean effective stress at the critical state (unit: kPa)
- \(p_\mathrm{c}\) :
-
Confining pressure (unit: kPa)
- \(p_\mathrm{a}\) :
-
Atmospheric pressure (unit: kPa)
- q :
-
Deviatoric stress (unit: kPa)
- \(q_{\mathrm{cs}}\) :
-
Critical state deviatoric stress (unit: kPa)
- \(\eta \) :
-
Stress ratio
- \(\varepsilon _\mathrm{a}\) :
-
Axial strain (unit: %)
- \(\varepsilon _\mathrm{v}\) :
-
Volumetric strain (unit: %)
- \(\hbox {d}\varepsilon _{v}^{p}\) :
-
Plastic volumetric strain increment (unit: %)
- \(\hbox {d}\varepsilon _{s}^{p}\) :
-
Plastic shear strain increment (unit: %)
- \(\hbox {d}\varepsilon _{v}\) :
-
Total volumetric strain increment (unit: %)
- \(\hbox {d}\varepsilon _{s}\) :
-
Total shear strain increment (unit: %)
- \(M_{\mathrm{cs}}\) :
-
Critical state stress ratio
- \(M_{\mathrm{cs0}},\quad \chi _\mathrm{M}^\mathrm{b}\) and \(k_\mathrm{M}\) :
-
Material constants of CSL in the \(q{-}{p}'\) space
- e :
-
Current void ratio
- \(e_{0}\) :
-
Initial void ratio
- \(e_{\mathrm{cs}}\) :
-
Critical state void ratio
- \(e_{\mathrm{cs0}} ,\lambda _{\mathrm{cs}}\) and \(\xi \) :
-
Material constants of CSL in the \(e{-}{p}'\) space
- \(e_{\mathrm{cs}0}^{0}\) and \(\chi _{\mathrm{e}}^{\mathrm{b}}\) :
-
Material constants in relation to \(e_{\mathrm{cs0}}\)
- \(\lambda _{\mathrm{cs}}^{0}\) and \(\chi _{\lambda }^\mathrm{b}\) :
-
Material constants in relation to \(\lambda _{\mathrm{cs}}\)
- d :
-
Dilatancy
- \(d_{\max }\) :
-
Maximum dilatancy
- \(\psi \) :
-
State parameter
- \(\varphi _{\mathrm{cs}}\) :
-
Critical state friction angle (unit: \(^\circ \))
- \(\varphi _{\mathrm{ps}}\) :
-
Peak state friction angle (unit: \(^\circ \))
- \(\varphi _{\mathrm{ex}}\) :
-
Excess friction angle (unit: \(^\circ \))
- \(\left( {{{\sigma }'_{1} }/{{\sigma }'_{3}}}\right) _{\mathrm{const}}\) :
-
Value of \(\left( {{{\sigma }'_{1}}/{{\sigma }'_{3}}} \right) \) at a constant volumetric strain
- \(\left( {{{\sigma }'_{1}}/{{\sigma }'_{3}}}\right) _{\max }\) :
-
Maximum value of \(\left( {{{\sigma }'_{1}}/{{\sigma }'_{3}}}\right) \)
- \(\chi _{\upvarphi }\) and \(\psi _\mathrm{a}\) :
-
Material constants in relation to \(\varphi _{\mathrm{ex}}\) (unit of \(\chi _{\upvarphi }\): \(^\circ \))
- \(d_{0}\) and \(\chi _\mathrm{d}\) :
-
Material constants in relation to \(d_{\max }\)
- \(\theta _{\max }\) :
-
Maximum dilatancy angle (unit: \(^\circ \))
- \(\theta _{0}\) and \(\chi _{\uptheta }\) :
-
Material constants in relation to \(\theta _{\max }\) (unit: \(^\circ \))
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Acknowledgments
The authors acknowledge the financial support from the 111 Project (Grant No. B13024), the National Natural Science Foundation of China (Grant No. 51509024) and the Fundamental Research Funds for the Central Universities (Grant No. 106112015CDJXY200008).
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**ao, Y., Sun, Y., Liu, H. et al. Critical state behaviors of a coarse granular soil under generalized stress conditions. Granular Matter 18, 17 (2016). https://doi.org/10.1007/s10035-016-0623-3
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DOI: https://doi.org/10.1007/s10035-016-0623-3