The stress–strain behaviour and critical state parameters of an unsaturated granular fill material under different suctions

Abstract

In mountainous western China, a large number of granular materials are used as construction materials in high-fill embankments. These granular fill materials in the embankments are typically unsaturated owing to the arid and semi-arid climates in this area. However, previous studies on unsaturated soils primarily focus on fine-grained soils. In this study, a series of triaxial tests were performed to determine the critical state parameters of a granular fill material in q–$\bar{p}$, v–ln $\bar{p}$, and v_w–ln $\bar{p}$ planes. An upgraded double-cell triaxial system (DCTS) was used in net confining pressures ranging from 0 to 450 kPa and matric suctions ranging from 0 to 160 kPa. This study demonstrates the good performance of the upgraded DCTS in unsaturated soil testing. Experimental results show that the critical state lines are almost parallel to those of saturated soil in the q–$\bar{p}$ plane, suggesting that the friction angle is independent of suction. The total cohesion and hence the shear strength increase with suction. In the v–ln $\bar{p}$ plane, both the intercept and slope of the critical state line increase with suction. Finally, it is observed that the intercept and slope decrease with increasing suction in the v_w–ln $\bar{p}$ plane.

Appendix: Mathematical formulations used in this study

1.1 Stress variables

Two stress variables adopted are expressed as follows, including net mean stress ($\bar{p}$) and matric suction (s):

$$ \bar{p} = p{-}u_{\text{a}} = \frac{{\sigma_{1} + \sigma_{2} + \sigma_{3} }}{3}{-}u_{\text{a}} $$

(1)

$$ s = u_{\text{a}} {-}u_{\text{w}} $$

(2)

where $\sigma_{1}$, $\sigma_{2}$, and $\sigma_{3}$ are principal stresses and u_a and u_w are pore-air pressure and pore-water pressure, respectively.

1.2 WRC model

A smooth and closed-form model proposed by van Genuchten [49] was used to fit the experimental data:

$$ S_{\text{r}} = \frac{1}{{[1 + (\psi /a)^{n} ]^{m} }} $$

(3)

where S_r denotes the degree of saturation, ψ denotes soil suction, and a, n, and m denote three curve-fitting parameters.

1.3 Isotropic compression stage

An equation was proposed by Thomas [45] to describe the excess pore-water pressure produced by the ramped consolidation with increasing total stress in a saturated soil specimen. The equation was used to derive the loading rate in isotropic compression stage as follows:

$$ R_{\text{L}} = \frac{{2u_{\text{ex}} c_{\text{v}} }}{{h^{2} }} $$

(4)

where R_L denotes the loading rate, h denotes drainage path length, c_v denotes coefficient of consolidation, and u_ex denotes excess pore-water pressure which was assumed as 2 kPa.

1.4 Critical state framework for saturated soils

The unique relationship is defined for the CSL in critical state framework. The equations for CSL proposed by Schofield and Wroth [32] are listed as follows:

$$ q = M_{\text{s}} p^{{\prime }} $$

(5)

$$ \nu = \varGamma_{\text{s}} {-}\lambda_{\text{s}} \ln p^{{\prime }} $$

(6)

$$ M_{\text{s}} = \frac{{6\sin \varphi^{{\prime }} }}{{3{-}\sin \varphi^{{\prime }} }} $$

(7)

where q denotes deviatoric stress, p′ denotes mean effective stress, M_s denotes slope of CSL in the q–p′ plane, Г_s denotes intercept of CSL at 1 kPa in the v–ln p′ plane, λ_s denotes slope of CSL in the v–ln p′ plane, and φ′ denotes angle of effective friction angle.

Additionally, Eqs. (8) and (9) are used to describe the compression and swelling behaviour of soil [32]:

$$ \nu = N_{\text{s}} {-}\lambda_{\text{s}} \ln p^{{\prime }} $$

(8)

$$ \nu = \nu_{\text{k}} {-}\kappa_{\text{s}} \ln p^{{\prime }} $$

(9)

where N_s denotes intercept of normal compression line (NCL) at 1 kPa in the v–ln p′ plane, λ_s denotes slope of NCL in the v–ln p′ plane, ν_κ denotes intercept of unloading/reloading curve at 1 kPa in the v–ln p′ plane, and κ_s denotes slope of unloading/reloading curve in the v–ln p′ plane.

1.5 Suction-based framework for unsaturated soils

Easy-to-understand linear equations for critical states proposed by Wheeler and Sivakumar [56] are listed as follows:

$$ q = M(s)\bar{p} + \mu (s) $$

(10)

$$ \nu = \varGamma (s) - \psi (s)\ln \left( {\frac{{\bar{p}}}{{p_{\text{at}} }}} \right) $$

(11)

$$ \nu_{\text{w}} = A(s) - B(s)\ln \left( {\frac{{\bar{p}}}{{p_{\text{at}} }}} \right) $$

(12)

where the parameters M(s) and μ(s) are the slopes and the intercepts of critical state lines in the q–$\bar{p}$ plane, respectively. Γ(s) and A(s) are the intercepts of critical state lines at $\bar{p}$ = 1 kPa in the v–ln $\bar{p}$ and v_w–ln $\bar{p}$ planes, respectively. ψ(s) and B(s) are slopes of critical state lines in the v–ln $\bar{p}$ and v_w–ln $\bar{p}$ planes, respectively. ν_w is specific water volume (ν_w = 1 + S_re), and p_at is the atmospheric pressure, taken as 100 kPa [56].

1.6 S_r-based framework for unsaturated soils

A degree-of-saturation-dependent framework proposed by Toll [46] and Toll and Ong [47] is listed as follows to model the critical state stress ratios for unsaturated soils:

$$ q = M_{\text{a}} (p - u_{\text{a}} ) + M_{\text{b}} (u_{\text{a}} - u_{\text{w}} ) $$

(13)

$$ \frac{{M_{\text{a}} }}{{M_{\text{s}} }} = \left( {\frac{{M_{\text{a}} }}{{M_{\text{s}} }}} \right)_{\hbox{max} } + \left[ {\left( {\frac{{M_{\text{a}} }}{{M_{\text{s}} }}} \right)_{\hbox{max} } - 1} \right]\left( {\frac{{S_{r} - S_{r2} }}{{S_{r1} - S_{r2} }}} \right)^{{k_{a} }} $$

(14)

$$ \frac{{M_{\text{b}} }}{{M_{\text{s}} }} = \left( {\frac{{S_{r} - S_{r2} }}{{S_{r1} - S_{r2} }}} \right)^{{k_{b} }} $$

(15)

where M_a and M_b are the stress ratios, which can define the shear strength arising from net mean stress and matric suction, respectively. S_r1 is the degree of saturation at full saturation (first reference state), S_r2 is the degree of saturation at residual suction (second reference state) [48], and parameters k_a and k_b are defined to provide a degree of curvature for the function between the two reference states.

The equations of critical state compressibilities for unsaturated soils are shown as follows:

$$ \varGamma_{\text{ab}} = 1 + \frac{{\varGamma_{\text{s}} - 1}}{{S_{\text{r}} }} $$

(16)

$$ \nu = \varGamma_{\text{ab}} - \lambda_{\text{a}} \ln (p - u_{\text{a}} ) - \lambda_{\text{b}} \ln (u_{\text{a}} - u_{\text{w}} ) $$

(17)

$$ \frac{{\lambda_{\text{a}} }}{{\lambda_{\text{s}} }} = \left( {\frac{{\lambda_{\text{a}} }}{{\lambda_{\text{s}} }}} \right)_{\hbox{max} } + \left[ {\left( {\frac{{\lambda_{\text{a}} }}{{\lambda_{\text{s}} }}} \right)_{\hbox{max} } - 1} \right]\left( {\frac{{S_{r} - S_{r2} }}{{S_{r1} - S_{r2} }}} \right)^{{k_{a} }} $$

(18)

$$ \frac{{\lambda_{b} }}{{\lambda_{\text{s}} }} = \left( {\frac{{S_{r} - S_{r2} }}{{S_{r1} - S_{r2} }}} \right)^{{k_{b} }} $$

(19)

where Γ_ab is a parameter related to Γ_s and S_r, λ_a and λ_b are functions of the degree of saturation and soil fabric, and S_r1, S_r2, k_a, and k_b are the parameters similar to those used for stress ratios.